Editing Glossary of ring theory

{{Short description|none}}
[[Ring theory]] is the branch of [[mathematics]] in which [[ring (mathematics)|rings]] are studied: that is, structures supporting both an [[addition]] and a [[multiplication]] operation. This is a glossary of some terms of the subject.

For the items in commutative algebra (the theory of commutative rings), see ''[[Glossary of commutative algebra]]''. For ring-theoretic concepts in the language of modules, see also ''[[Glossary of module theory]]''.

For specific types of algebras, see also: ''[[Glossary of field theory]]'' and ''[[Glossary of Lie groups and Lie algebras]]''. Since, currently, there is no glossary on not-necessarily-associative algebra structures in general, this glossary includes some concepts that do not need associativity; e.g., a derivation.

{{Compact ToC|short1|x=[[#XYZ|XYZ]]|y=|z=|seealso=yes|refs=yes}}

== A ==
{{glossary}}

{{term|1=Amitsur complex}}
{{defn|1=The [[Amitsur complex]] of a ring homomorphism is a cochain complex that measures the extent in which the ring homomorphism fails to be [[faithfully flat ring homomorphism|faithfully flat]].}}

{{term|1=Artinian}}
{{defn|1=A left [[Artinian ring]] is a ring satisfying the [[descending chain condition]] for left ideals; a right Artinian ring is one satisfying the descending chain condition for right ideals. If a ring is both left and right Artinian, it is called ''Artinian''. Artinian rings are Noetherian rings.}}

{{term|1=associate}}
{{defn|1=In a commutative ring, an element ''a'' is called an [[Divisibility (ring theory)|associate]] of an element ''b'' if ''a'' divides ''b'' and ''b'' divides ''a''.}}

{{term|1=automorphism}}
{{defn|1=A [[ring automorphism]] is a ring isomorphism between the same ring; in other words, it is a unit element of the endomorphism ring of the ring that is multiplicative and preserves the multiplicative identity.}}
{{defn|1=An [[algebra automorphism]] over a commutative ring ''R'' is an algebra isomorphism between the same algebra; it is a ring automorphism that is also ''R''-linear.}}

{{term|1=Azumaya}}
{{defn|1=An [[Azumaya algebra]] is a generalization of a central simple algebra to a non-field base ring.}}

{{glossary end}}

== B ==
{{glossary}}

{{term|1=bidimension}}
{{defn|1=The [[bidimension of an associative algebra]] ''A'' over a commutative ring ''R'' is the projective dimension of ''A'' as an {{nowrap|(''A''<sup>op</sup> ⊗<sub>''R''</sub> ''A'')}}-module. For example, an algebra has bidimension zero if and only if it is separable.}}

{{term|1=boolean}}
{{defn|1=A [[boolean ring]] is a ring in which every element is multiplicatively [[idempotent (ring theory)|idempotent]].}}

{{term|1=Brauer}}
{{defn|1=The [[Brauer group]] of a field is an abelian group consisting of all equivalence classes of central simple algebras over the field.}}

{{glossary end}}

== C ==
{{glossary}}
{{term|1=category}}
{{defn|1=The [[category of rings]] is a category where the objects are (all) the rings and where the morphisms are (all) the ring homomorphisms.}}

{{term|1=centre}}
{{defn|no=1|1=An element ''r'' of a ring ''R'' is ''central'' if {{nowrap|1=''xr'' = ''rx''}} for all ''x'' in ''R''. The set of all central elements forms a [[subring]] of ''R'', known as the [[center (ring theory)|centre]] of ''R''.}}
{{defn|no=2|1=A [[central algebra]] is an associative algebra over the centre.}}
{{defn|no=3|1=A [[central simple algebra]] is a central algebra that is also a simple ring.}}

{{term|1=centralizer}}
{{defn|no=1|The [[centralizer (ring theory)|centralizer]] of a subset ''S'' of a ring is the subring of the ring consisting of the elements commuting with the elements of ''S''. For example, the centralizer of the ring itself is the centre of the ring.}}
{{defn|no=2|The [[double centralizer]] of a set is the centralizer of the centralizer of the set. Cf. [[double centralizer theorem]].}}

{{term|1=characteristic}}
{{defn|no=1|1=The [[Characteristic (algebra)|characteristic]] of a ring is the smallest positive integer ''n'' satisfying ''nx'' = 0 for all elements ''x'' of the ring, if such an ''n'' exists. Otherwise, the characteristic is 0.}}
{{defn|no=2|1=The [[characteristic subring]] of ''R'' is the smallest subring (i.e., the unique minimal subring). It is necessary the image of the unique ring homomorphism {{nowrap|'''Z''' → ''R''}} and thus is isomorphic to '''Z'''/''n'' where ''n'' is the characteristic of ''R''.}}

{{term|1=change}}
{{defn|A [[change of rings]] is a functor (between appropriate categories) induced by a ring homomorphism.}}

{{term|1=Clifford algebra}}
{{defn|1=A [[Clifford algebra]] is a certain associative algebra that is useful in geometry and physics.}}

{{term|coherent}}
{{defn|1=A left [[coherent ring]] is a ring such that every finitely generated left ideal of it is a finitely presented module; in other words, it is [[coherent module|coherent]] as a left module over itself.}}

{{term|1=commutative}}
{{defn|no=1|1=A ring ''R'' is [[commutative ring|commutative]] if the multiplication is commutative, i.e. {{nowrap|1=''rs'' = ''sr''}} for all {{nowrap|''r'',''s'' ∈ ''R''}}.}}
{{defn|no=2|1=A ring ''R'' is [[skew-commutative ring]] if {{nowrap|1=''xy'' = (−1)<sup>''ε''(''x'')''ε''(''y'')</sup>''yx''}}, where ''ε''(''x'') denotes the parity of an element ''x''.}}
{{defn|no=3|1=A commutative algebra is an associative algebra that is a commutative ring.}}
{{defn|no=4|1=[[Commutative algebra]] is the theory of commutative rings.}}

{{glossary end}}

== D ==
{{glossary}}

{{term|1=derivation}}
{{defn|no=1|1=A [[derivation of an algebra|derivation]] of a possibly-non-associative algebra ''A'' over a commutative ring ''R'' is an ''R''-linear endomorphism that satisfies the [[Product rule|Leibniz rule]].}}
{{defn|no=2|1=The [[derivation algebra]] of an algebra ''A'' is the subalgebra of the endomorphism algebra of ''A'' that consists of derivations.}}

{{term|1=differential}}
{{defn|1=A [[differential algebra]] is an algebra together with a derivation.}}

{{term|1=direct}}
{{defn|1=A [[direct product ring|direct product]] of a family of rings is a ring given by taking the [[cartesian product]] of the given rings and defining the algebraic operations component-wise.}}

{{term|1=divisor}}
{{defn|no=1|1=In an [[integral domain]] ''R'',{{clarify|Do we need to assume ''R'' is an integral domain?|date=January 2020}} an element ''a'' is called a [[Divisibility (ring theory)|divisor]] of the element ''b'' (and we say ''a'' ''divides'' ''b'') if there exists an element ''x'' in ''R'' with {{nowrap|1=''ax'' = ''b''}}.}}
{{defn|no=2|1=An element ''r'' of ''R'' is a ''left [[zero divisor]]'' if there exists a nonzero element ''x'' in ''R'' such that {{nowrap|1=''rx'' = 0}} and a ''right zero divisor'' or if there exists a nonzero element ''y'' in ''R'' such that {{nowrap|1=''yr'' = 0}}. An element ''r'' of ''R'' is a called a ''two-sided zero divisor'' if it is both a left zero divisor and a right zero divisor.}}

{{term|1=division}}
{{defn|1=A [[division ring]] or skew field is a ring in which every nonzero element is a unit and {{nowrap|1 ≠ 0}}.}}

{{term|1=domain}}
{{defn|1=A [[Domain (ring theory)|domain]] is a nonzero ring with no zero divisors except 0. For a historical reason, a commutative domain is called an [[integral domain]].}}

{{glossary end}}

== E ==
{{glossary}}

{{term|1=endomorphism}}
{{defn|1=An [[endomorphism ring]] is a ring formed by the [[endomorphism]]s of an object with additive structure; the multiplication is taken to be [[function composition]], while its addition is pointwise addition of the images.}}

{{term|1=enveloping algebra}}
{{defn|1=The (universal) [[enveloping algebra (disambiguation)|enveloping algebra]] ''E'' of a not-necessarily-associative algebra ''A'' is the associative algebra determined by ''A'' in some universal way. The best known example is the [[universal enveloping algebra]] of a Lie algebra.}}

{{term|1=extension}}
{{defn|1=A ring ''E'' is a [[ring extension]] of a ring ''R'' if ''R'' is a [[subring]] of ''E''.}}

{{term|1=exterior algebra}}
{{defn|1=The [[exterior algebra]] of a vector space or a module ''V'' is the quotient of the tensor algebra of ''V'' by the ideal generated by elements of the form {{nowrap|''x'' ⊗ ''x''}}.}}

{{glossary end}}

== F ==
{{glossary}}

{{term|1=field}}
{{defn|1=A [[Field (mathematics)|field]] is a commutative division ring; i.e., a nonzero ring in which each nonzero element is invertible.}}

{{term|1=filtered ring}}
{{defn|1=A [[filtered ring]] is a ring with a filtration.}}

{{term|1=finitely generated}}
{{defn|no=1|1=A left ideal ''I'' is ''[[finitely generated ideal|finitely generated]]'' if there exist finitely many elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} such that {{nowrap|1=''I'' = ''Ra''<sub>1</sub> + ... + ''Ra''<sub>''n''</sub>}}. A right ideal ''I'' is ''finitely generated'' if there exist finitely many elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} such that {{nowrap|1=''I'' = ''a''<sub>1</sub>''R'' + ... + ''a''<sub>''n''</sub>''R''}}. A two-sided ideal ''I'' is ''finitely generated'' if there exist finitely many elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} such that {{nowrap|1=''I'' = ''Ra''<sub>1</sub>''R'' + ... + ''Ra''<sub>''n''</sub>''R''}}.}}
{{defn|no=2|A '''[[finitely generated ring]]''' is a ring that is finitely generated as '''Z'''-algebra.}}

{{term|1=finitely presented}}
{{defn|1=A finitely presented algebra over a commutative ring ''R'' is a (commutative) [[associative algebra]] that is a [[quotient ring|quotient]] of a [[polynomial ring]] over ''R'' in finitely many variables by a [[finitely generated ideal]].<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.4.1}}</ref><!-- the non-commutative case should be discussed. -->}}

{{term|1=free}}
{{defn|no=1|1=A [[free ideal ring]] or a fir is a ring in which every right ideal is a free module of fixed rank.}}
{{defn|no=2|1=A semifir is a ring in which every finitely generated right ideal is a free module of fixed rank.}}
{{defn|no=3|1=The [[free product of associative algebras|free product]] of a family of associative is an associative algebra obtained, roughly, by the generators and the relations of the algebras in the family. The notion depends on which category of associative algebra is considered; for example, in the category of commutative rings, a free product is a tensor product.}}
{{defn|no=4|1=A [[free ring]] is a ring that is a [[free algebra]] over the integers.}}

{{glossary end}}

== G ==
{{glossary}}

{{term|1=graded}}
{{defn|1=A [[graded ring]] is a ring together with a grading or a graduation; i.e, it is a direct sum of additive subgroups with the multiplication that respects the grading. For example, a polynomial ring is a graded ring by degrees of polynomials.}}

{{term|1=generate}}
{{defn|1=An associative algebra ''A'' over a commutative ring ''R'' is said to be [[algebra generated by a set|generated]] by a subset ''S'' of ''A'' if the smallest subalgebra containing ''S'' is ''A'' itself and ''S'' is said to be the generating set of ''A''. If there is a finite generating set, ''A'' is said to be a [[finitely generated algebra]].}}

{{glossary end}}

== H ==
{{glossary}}

{{term|1=hereditary}}
{{defn|1=A ring is [[hereditary ring|left hereditary]] if its left ideals are all projective modules. Right hereditary rings are defined analogously.}}

{{glossary end}}

== I ==
{{glossary}}

{{term|1=ideal}}
{{defn|A [[left ideal]] ''I'' of ''R'' is an additive subgroup of ''R'' such that {{nowrap|''aI'' ⊆ ''I''}} for all {{nowrap|''a'' ∈ ''R''}}. A ''right ideal'' is a subgroup of ''R'' such that {{nowrap|''Ia'' ⊆ ''I''}} for all {{nowrap|''a'' ∈ ''R''}}. An ''ideal'' (sometimes called a ''two-sided ideal'' for emphasis) is a subgroup that is both a left ideal and a right ideal.}}

{{term|1=idempotent}}
{{defn|1=An element ''r'' of a ring is [[Idempotent element (ring theory)|idempotent]] if {{nowrap|1=''r''{{i sup|2}} = ''r''}}.}}

{{term|1=integral domain}}
{{defn|1="'''[[integral domain]]'''" or "'''entire ring'''" is another name for a [[commutative domain]]; i.e., a nonzero [[commutative ring]] with no [[zero divisor]]s except 0.}}

{{term|1=invariant}}
{{defn|1=A ring ''R'' has [[invariant basis number]] if ''R''<sup>''m''</sup> isomorphic to ''R''<sup>''n''</sup> as [[module (mathematics)|''R''-modules]] implies {{nowrap|1=''m'' = ''n''}}.}}

{{term|1=irreducible}}
{{defn|1=An element ''x'' of an integral domain is [[Irreducible element|irreducible]] if it is not a unit and for any elements ''a'' and ''b'' such that {{nowrap|1=''x'' = ''ab''}}, either ''a'' or ''b'' is a unit. Note that every prime element is irreducible, but not necessarily vice versa.}}

{{glossary end}}

== J ==
{{glossary}}

{{term|1=Jacobson}}
{{defn|no=1|1=The [[Jacobson radical]] of a ring is the intersection of all maximal left ideals.}}
{{defn|no=2|1=A [[Jacobson ring]] is a ring in which each prime ideal is an intersection of primitive ideals.}}

{{glossary end}}

== K ==
{{glossary}}

{{term|1=kernel}}
{{defn|1=The [[kernel (algebra)|kernel]] of a ring homomorphism of a ring homomorphism {{nowrap|''f'' : ''R'' → ''S''}} is the set of all elements ''x'' of ''R'' such that {{nowrap|1=''f''(''x'') = 0}}. Every ideal is the kernel of a ring homomorphism and vice versa.}}

{{term|1=Köthe}}
{{defn|1=[[Köthe's conjecture]] states that if a ring has a nonzero nil right ideal, then it has a nonzero nil ideal.}}

{{glossary end}}

== L ==
{{glossary}}

{{term|1=local}}
{{defn|no=1|1=A ring with a unique maximal left ideal is a [[local ring]]. These rings also have a unique maximal right ideal, and the left and the right unique maximal ideals coincide. Certain commutative rings can be embedded in local rings via [[localization of a ring|localization]] at a [[prime ideal]].}}
{{defn|no=2|1=A [[localization of a ring]] : For commutative rings, a technique to turn a given set of elements of a ring into units. It is named ''Localization'' because it can be used to make any given ring into a ''local'' ring. To localize a ring ''R'', take a multiplicatively closed subset ''S'' that contains no [[zero divisor]]s, and formally define their multiplicative inverses, which are then added into ''R''. Localization in noncommutative rings is more complicated, and has been in defined several different ways.}}

{{glossary end}}

== M ==
{{glossary}}

{{term|1=minimal and maximal}}
{{defn|no=1|1=A left ideal ''M'' of the ring ''R'' is a [[maximal ideal|maximal left ideal]] (resp. minimal left ideal) if it is maximal (resp. minimal) among proper (resp. nonzero) left ideals. Maximal (resp. minimal) right ideals are defined similarly.}}
{{defn|no=2|1=A [[maximal subring]] is a subring that is maximal among proper subrings. A "minimal subring" can be defined analogously; it is unique and is called the [[characteristic subring]].}}

{{term|1=matrix}}
{{defn|no=1|1=A [[matrix ring]] over a ring ''R'' is a ring whose elements are square matrices of fixed size with the entries in ''R''. The matrix ring or the full matrix ring of matrices over ''R'' is ''the'' matrix ring consisting of all square matrices of fixed size with the entries in ''R''. When the grammatical construction is not workable, the term "matrix ring" often refers to the "full" matrix ring when the context makes no confusion likely; for example, when one says a semsimple ring is a product of matrix rings of division rings, it is implicitly assumed that "matrix rings" refer to "full matrix rings". Every ring is (isomorphic to) the full matrix ring over itself.}}
{{defn|no=2|1=The [[ring of generic matrices]] is the ring consisting of square matrices with entries in formal variables.}}

{{term|1=monoid}}
{{defn|1=A [[monoid ring]].}}

{{term|1=Morita}}
{{defn|1=Two rings are said to be [[Morita equivalent]] if the [[category of modules]] over the one is equivalent to the category of modules over the other.}}

{{glossary end}}

== N ==
{{glossary}}

{{term|1=nearring}}
{{defn|1=A '''[[nearring]]''' is a structure that is a group under addition, a [[semigroup]] under multiplication, and whose multiplication distributes on the right over addition.}}

{{term|1=nil}}
{{defn|no=1|1=A [[nil ideal]] is an ideal consisting of nilpotent elements.}}
{{defn|no=2|1=The (Baer) [[nilradical of a ring|upper nil radical]] is the sum of all nil ideals.}}
{{defn|no=3|1=The (Baer) [[nilradical of a ring|lower nil radical]] is the intersection of all prime ideals. For a commutative ring, the upper nil radical and the lower nil radical coincide.}}

{{term|1=nilpotent}}
{{defn|no=1|An element ''r'' of ''R'' is [[nilpotent element|nilpotent]] if there exists a positive integer ''n'' such that {{nowrap|1=''r''{{i sup|''n''}} = 0}}.}}
{{defn|no=2|1=A [[nil ideal]] is an ideal whose elements are nilpotent elements.}}
{{defn|no=3|1=A [[nilpotent ideal]] is an ideal whose [[product of ideals|power]] ''I''<sup>''k''</sup> is {0} for some positive integer ''k''. Every nilpotent ideal is nil, but the converse is not true in general.}}
{{defn|no=4|1=The [[Nilradical of a ring|nilradical]] of a commutative ring is the ideal that consists of all nilpotent elements of the ring. It is equal to the intersection of all the ring's [[prime ideal]]s and is contained in, but in general not equal to, the ring's Jacobson radical.}}

{{term|Noetherian}}
{{defn|1=A left [[Noetherian ring]] is a ring satisfying the [[ascending chain condition]] for left ideals. A ''right Noetherian'' is defined similarly and a ring that is both left and right Noetherian is ''Noetherian''. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for right Noetherian rings.}}

{{term|1=null}}
{{defn|1='''null ring''': See {{gli|rng of square zero}}.}}

{{glossary end}}

== O ==
{{glossary}}

{{term|1=opposite}}
{{defn|1=Given a ring ''R'', its [[opposite ring]] ''R''<sup>op</sup> has the same underlying set as ''R'', the addition operation is defined as in ''R'', but the product of ''s'' and ''r'' in ''R''<sup>op</sup> is ''rs'', while the product is ''sr'' in ''R''.}}

{{term|1=order}}
{{defn|1=An [[order (ring theory)|order]] of an algebra is (roughly) a subalgebra that is also a full lattice.}}

{{term|1=Ore}}
{{defn|1=A left [[Ore domain]] is a (non-commutative) domain for which the set of non-zero elements satisfies the left Ore condition. A right Ore domain is defined similarly.}}

{{glossary end}}

== P ==
{{glossary}}

{{term|1=perfect}}
{{defn|1=A ''left [[perfect ring]]'' is one satisfying the [[descending chain condition]] on ''right'' principal ideals. They are also characterized as rings whose flat left modules are all projective modules. Right perfect rings are defined analogously. Artinian rings are perfect.}}

{{term|1=polynomial}}
{{defn|no=1|1=A [[polynomial ring]] over a commutative ring ''R'' is a commutative ring consisting of all the polynomials in the specified variables with coefficients in&nbsp;''R''.}}
{{defn|no=2|A [[Polynomial ring#Differential and skew-polynomial rings|skew polynomial ring]]
: Given a ring ''R'' and an endomorphism {{nowrap|''σ'' ∈ End(''R'')}} of ''R''. The skew polynomial ring {{nowrap|''R''[''x''; ''σ'']}} is defined to be the set {{nowrap|{{mset|''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ''a''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ... + ''a''<sub>1</sub>''x'' + ''a''<sub>0</sub> {{!}} ''n'' ∈ '''N''', ''a''<sub>''n''</sub>, ''a''<sub>''n''−1</sub>, ..., ''a''<sub>1</sub>, ''a''<sub>0</sub> ∈ ''R''}}}}, with addition defined as usual, and multiplication defined by the relation {{nowrap|1=''xa'' = ''σ''(''a'')''x'' ∀''a'' ∈ ''R''}}.}}

{{term|1=prime}}
{{defn|no=1|1=An element ''x'' of an integral domain is a [[prime element]] if it is not zero and not a unit and whenever ''x'' divides a product ''ab'', ''x'' divides ''a'' or ''x'' divides ''b''.}}
{{defn|no=2|1=An ideal ''P'' in a [[commutative ring]] ''R'' is [[prime ideal|prime]] if {{nowrap|''P'' ≠ ''R''}} and if for all ''a'' and ''b'' in ''R'' with ''ab'' in ''P'', we have ''a'' in ''P'' or ''b'' in ''P''. Every maximal ideal in a commutative ring is prime.}}
{{defn|no=3|1=An ideal ''P'' in a (not necessarily commutative) ring ''R'' is prime if {{nowrap|''P'' ≠ ''R''}} and for all ideals ''A'' and ''B'' of ''R'', {{nowrap|''AB'' ⊆ ''P''}} implies {{nowrap|''A'' ⊆ ''P''}} or {{nowrap|''B'' ⊆ ''P''}}.  This extends the definition for commutative rings.}}
{{defn|no=4|'''[[prime ring]]''' : A [[zero ring|nonzero ring]] ''R'' is called a ''prime ring'' if for any two elements ''a'' and ''b'' of ''R'' with {{nowrap|1=''aRb'' = 0}}, we have either {{nowrap|1=''a'' = 0}} or {{nowrap|1=''b'' = 0}}. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every [[simple ring]] and every [[domain (ring theory)|domain]] is a prime ring.}}

{{term|1=primitive}}
{{defn|no=1|1=A ''left [[primitive ring]]'' is a ring that has a [[faithful module|faithful]] [[simple module|simple]] [[module (mathematics)|left ''R''-module]]. Every [[simple ring]] is primitive.  Primitive rings are [[prime ring|prime]].}}
{{defn|no=2|1=An ideal ''I'' of a ring ''R'' is said to be [[primitive ideal|primitive]] if ''R''/''I'' is primitive.}}

{{term|1=principal}}
{{defn|1=A '''[[principal ideal]]''' : A ''principal left ideal'' in a ring ''R'' is a left ideal of the form ''Ra'' for some element ''a'' of ''R''. A ''principal right ideal'' is a right ideal of the form ''aR'' for some element ''a'' of ''R''. A ''principal ideal'' is a two-sided ideal of the form ''RaR'' for some element ''a'' of ''R''.}}

{{term|1=principal}}
{{defn|no=1|A [[principal ideal domain]] is an integral domain in which every ideal is principal.}}
{{defn|no=2|A [[principal ideal ring]] is a ring in which every ideal is principal.}}

{{glossary end}}

== Q ==
{{glossary}}

{{term|1=quasi-Frobenius}}
{{defn|1='''[[quasi-Frobenius ring]]''' : a special type of Artinian ring that is also a [[self-injective ring]] on both sides. Every semisimple ring is quasi-Frobenius.}}

{{defn|1='''[[quotient ring]]''' or '''factor ring''' : Given a ring ''R'' and an ideal ''I'' of ''R'', the ''quotient ring'' is the ring formed by the set ''R''/''I'' of [[coset]]s {{nowrap|{{mset|''a'' + ''I'' : ''a'' ∈ ''R''}}}} together with the operations {{nowrap|1=(''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + ''I''}} and {{nowrap|1=(''a'' + ''I'')(''b'' + ''I'') = ''ab'' + ''I''}}. The relationship between ideals, homomorphisms, and factor rings is summed up in the [[fundamental theorem on homomorphisms]].}}

{{glossary end}}

== R ==
{{glossary}}

{{term|1=radical}}
{{defn|1=The [[radical of an ideal]] ''I'' in a [[commutative ring]] consists of all those ring elements a power of which lies in ''I''. It is equal to the intersection of all prime ideals containing ''I''.}}

{{term|1=ring}}
{{defn|no=1|1=A [[Set (mathematics)|set]] ''R'' with two [[binary operation]]s, usually called addition (+) and multiplication (×), such that ''R'' is an [[abelian group]] under addition, ''R'' is a [[monoid]] under multiplication, and multiplication is both left and right [[distributive property|distributive]] over addition. Rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1. (''Warning'': some books, especially older books, use the term "ring" to mean what here will be called a [[rng (algebra)|rng]]; i.e., they do not require a ring to have a multiplicative identity.)}}
{{defn|no=2|1=A '''[[ring homomorphism]]''' : A [[function (mathematics)|function]] {{nowrap|''f'' : ''R'' → ''S''}} between rings {{nowrap|(''R'', +, ∗)}} and {{nowrap|(''S'', ⊕, ×)}} is a ''ring homomorphism'' if it satisfies
:: ''f''(''a'' + ''b'') = ''f''(''a'') ⊕ ''f''(''b'')
:: ''f''(''a'' ∗ ''b'') = ''f''(''a'') × ''f''(''b'')
:: ''f''(1) = 1
:for all elements ''a'' and ''b'' of ''R''.}}
{{defn|no=3|'''[[ring isomorphism]]''' : A ring homomorphism that is [[bijective]] is a ''ring isomorphism''. The inverse of a ring isomorphism is also a ring isomorphism. Two rings are ''isomorphic'' if there exists a ring isomorphism between them. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.}}

{{term|1=rng}}
{{defn|no=1|1=A '''[[rng (algebra)|rng]]''' is a [[Set (mathematics)|set]] ''R'' with two [[binary operation]]s, usually called addition (+) and multiplication (×), such that {{nowrap|(''R'', +)}} is an [[abelian group]], {{nowrap|(''R'', ×)}} is a [[semigroup]], and multiplication is both left and right [[distributive property|distributive]] over addition.  A rng that has an '''i'''dentity element is a "r'''i'''ng".}}
{{defn|no=2|1=A '''[[rng (algebra)#Rng of square zero|rng of square zero]]''' is a [[rng (algebra)|rng]] in which {{nowrap|1=''xy'' = 0}} for all ''x'' and ''y''.}}

{{glossary end}}

== S ==
{{glossary}}

{{term|1=self-injective}}
{{defn|1=A ring ''R'' is ''left [[self-injective ring|self-injective]]'' if the module <sub>''R''</sub>''R'' is an [[injective module]]. While rings with unity are always projective as modules, they are not always injective as modules.}}

{{term|1=semiperfect}}
{{defn|1=A [[semiperfect ring]] is a ring ''R'' such that, for the Jacobson radical J(''R'') of ''R'', (1) ''R''/J(''R'') is semisimple and (2) idempotents lift modulo J(''R'').}}

{{term|1=semiprimary}}
{{defn|1=A [[semiprimary ring]] is a ring ''R'' such that, for the Jacobson radical J(''R'') of ''R'', (1) ''R''/J(''R'') is semisimple and (2) J(''R'') is a [[nilpotent ideal]].}}

{{term|1=semiprime}}
{{defn|no=1|1=A [[semiprime ring]] is a ring where the only [[nilpotent ideal]] is the trivial ideal {{mset|0}}. A commutative ring is semiprime if and only if it is reduced.}}
{{defn|no=2|1=An ideal ''I'' of a ring ''R'' is [[semiprime ideal|semiprime]] if for any ideal ''A'' of ''R'', {{nowrap|''A''<sup>''n''</sup> ⊆ ''I''}} implies {{nowrap|''A'' ⊆ ''I''}}. Equivalently, ''I'' is semiprime if and only if ''R''/''I'' is a semiprime ring.}}

{{term|1=semiprimitive}}
{{defn|1=A [[semiprimitive ring]] or Jacobson semisimple ring is a ring whose [[Jacobson radical]] is zero. Von Neumann regular rings and primitive rings are semiprimitive, however quasi-Frobenius rings and local rings are usually not semiprimitive.}}

{{term|1=semiring}}
{{defn|1=A '''[[semiring]]''' : An algebraic structure satisfying the same properties as a ring, except that addition need only be an abelian [[monoid]] operation, rather than an abelian group operation. That is, elements in a semiring need not have additive inverses.}}

{{term|1=semisimple}}
{{defn|1=A [[semisimple ring]] is an Artinian ring ''R'' that is a finite product of simple Artinian rings; in other words, it is a [[semisimple module|semisimple]] left ''R''-module.}}

{{term|1=separable}}
{{defn|1=A [[separable algebra]] is an associative algebra whose tensor-square admits a [[separability idempotent]].}}

{{term|1=serial}}
{{defn|1=A right [[serial ring]] is a ring that is a right serial module over itself.}}

{{term|1=Severi–Brauer}}
{{defn|1=The [[Severi–Brauer variety]] is an algebraic variety associated to a given central simple algebra.}}

{{term|1=simple}}
{{defn|no=1|1=A [[simple ring]] is a non-zero ring that only has trivial two-sided ideals (the zero ideal, the ring itself, and no more) is a ''simple ring''.}}
{{defn|no=2|1=A [[simple algebra]] is an associative algebra that is a simple ring.}}

{{term|1=singular submodule}}
{{defn|1=The right (resp. left) ''R''-[[module (mathematics)|module]] ''M'' has a [[singular submodule]] if it consists of elements whose [[annihilator (ring theory)|annihilator]]s are [[essential submodule|essential]] right (resp. left) [[ideal (ring theory)|ideal]]s in ''R''.  In set notation it is usually denoted as {{nowrap|1={{mathcal|{{big|Z}}}}(''M'') = {{mset|''m'' ∈ ''M'' {{pipe}} ann(''m'') ⊆<sub>''e''</sub> ''R''}}}}.}}

{{term|1=subring}}
{{defn|1=A [[subring]] is a subset ''S'' of the ring {{nowrap|(''R'', +, ×)}} that remains a ring when + and × are restricted to ''S'' and contains the multiplicative identity 1 of ''R''.}}

{{term|1=symmetric algebra}}
{{defn|no=1|1=The [[symmetric algebra]] of a vector space or a module ''V'' is the quotient of the tensor algebra of ''V'' by the ideal generated by elements of the form {{nowrap|''x'' ⊗ ''y'' − ''y'' ⊗ ''x''}}.}}
{{defn|no=2|1=The [[graded-symmetric algebra]] of a vector space or a module ''V'' is a variant of the symmetric algebra that is constructed by taking grading into account.}}

{{term|1=Sylvester domain}}
{{defn|1=A [[Sylvester domain]] is a ring in which [[Sylvester's law of nullity]] holds.}}

{{glossary end}}

== T ==
{{glossary}}

{{term|1=tensor}}
{{defn|1=The [[tensor product algebra]] of associative algebras is the tensor product of the algebras as the modules with component multiplication}}
{{defn|1=The [[tensor algebra]] of a vector space or a module ''V'' is the direct sum of all tensor powers ''V''<sup>⊗''n''</sup> with the multiplication given by tensor product.}}

{{term|1=trivial}}
{{defn|no=1|1=A trivial ideal is either the zero or the unit ideal.}}
{{defn|no=2|1=The [[trivial ring]] or [[zero ring]] is the ring consisting of a single element {{nowrap|1=0 = 1}}.}}

{{glossary end}}

== U ==
{{glossary}}

{{term|1=unit}}
{{defn|1='''[[Unit (ring theory)|unit]]''' or '''invertible element''' : An element ''r'' of the ring ''R'' is a ''unit'' if there exists an element ''r''{{i sup|−1}} such that {{nowrap|1=''rr''{{i sup|−1}} = ''r''{{i sup|−1}}''r'' = 1}}. This element ''r''{{i sup|−1}} is uniquely determined by ''r'' and is called the ''multiplicative inverse'' of ''r''. The set of units forms a [[group (mathematics)|group]] under multiplication.}}

{{term|1=unity}}
{{defn|1=The term "unity" is another name for the multiplicative identity.}}

{{term|1=unique}}
{{defn|1=A '''[[unique factorization domain]]''' or '''factorial ring''' is an integral domain ''R'' in which every non-zero non-[[Unit (ring theory)|unit]] element can be written as a product of [[prime element]]s of ''R''.}}

{{term|1=uniserial}}
{{defn|1=A right [[uniserial ring]] is a ring that is a right uniserial module over itself. A commutative uniserial ring is also called a [[valuation ring]].}}

{{glossary end}}

== V ==
{{glossary}}

{{term|1=von Neumann regular element}}
{{defn|no=1|1='''[[von Neumann regular element]]''' : An element ''r'' of a ring ''R'' is ''von Neumann regular'' if there exists an element ''x'' of ''R'' such that {{nowrap|1=''r'' = ''rxr''}}.}}
{{defn|no=2|A '''[[von Neumann regular ring]]''': A ring for which each element ''a'' can be expressed as {{nowrap|1=''a'' = ''axa''}} for another element ''x'' in the ring. Semisimple rings are von Neumann regular.}}

{{glossary end}}

== W ==
{{glossary}}

{{term|1=Wedderburn–Artin theorem}}
{{defn|1=The [[Wedderburn–Artin theorem]] states that a semisimple ring is a finite product of (full) matrix rings over division rings.}}

{{glossary end}}
{{anchor|XYZ}}

== Z ==
{{glossary}}

{{term|1=zero}}
{{defn|1=A '''[[zero ring]]''': The ring consisting only of a single element {{nowrap|1=0 = 1}}, also called the [[trivial ring]]. Sometimes "zero ring" is used in an alternative sense to mean [[rng (algebra)#Rng of square zero|rng of square zero]].}}

{{glossary end}}

== See also ==
* [[Glossary of module theory]]

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation   |last1=Anderson |first1=Frank W.  |last2=Fuller |first2=Kent R.   |year=1992   |title=Rings and categories of modules   |series=[[Graduate Texts in Mathematics]]   |volume=13   |edition=2   |publisher=Springer-Verlag  |place=New York   |pages=x+376   |isbn=0-387-97845-3 |mr=1245487 |doi=10.1007/978-1-4612-4418-9 }}
* {{cite web |last1=Artin |first1=Michael |year=1999 |title=Noncommutative Rings |url=http://math.mit.edu/~etingof/artinnotes.pdf }}
* {{EGA | book=IV-1}}
* {{citation |last1=Jacobson |first1=Nathan |year=1956 |title=Structure of Rings |series=Colloquium Publications |volume=37 |publisher=American Mathematical Society |isbn=978-0-8218-1037-8 |url=https://books.google.com/books?id=KwviDgAAQBAJ }}
* {{citation |last1=Jacobson |first1=Nathan |year=2009 |title=Basic Algebra 1 |edition=2nd |publisher=Dover }}
* {{citation |last1=Jacobson |first1=Nathan |year=2009 |title=Basic Algebra 2 |edition=2nd |publisher=Dover }}
{{refend}}

{{DEFAULTSORT:Glossary Of Ring Theory}}
[[Category:Glossaries of mathematics|Ring theory]]
[[Category:Ring theory| ]]
[[Category:Wikipedia glossaries using description lists]]