Editing Glossary of ring theory (section)

== 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}}