Editing Glossary of ring theory (section)

== 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}}