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{{short description|Theorem about commutative rings and subrings}} .../math> of <math>A</math>, <math>\operatorname{length}_B (B/ Q B)</math> is finite and moreover ...

| title = Co-groups and co-rings in categories of associative rings ...]] <math>p</math>, <math>\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}</math> is a [[finite field]] with <math>p</math> elements and hence has characteristic <math>p</ ...

| title=Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay | pages=115–175}}</ref> states that rings of invariants of linearly [[reductive group]]s acting on [[regular ring]]s ...

In rings with unity it is always the case that <math>\mathcal{Z}(R_R)\subsetneq R\,< *If ''R'' is a right [[injective module#Self-injective rings|self-injective ring]], then <math>\mathcal{Z}(R_R)=J(R)\,</math>, where J(' ...

...se <math>f</math> is called a '''finite morphism''' if <math>A</math> is a finite <math>R</math>-algebra.<ref>{{cite book|url=https://www.crcpress.com/Introd ...ly generated algebra#finite algebras vs algebras of finite type|algebra of finite type]]. ...

...ons of quasi-Frobenius rings include '''QF-1''', '''QF-2''' and '''QF-3''' rings. ...ned by [[Georg Frobenius]]. A partial list of pioneers in quasi-Frobenius rings includes [[Richard Brauer|R. Brauer]], [[Kiiti Morita|K. Morita]], [[Tadash ...

...esentation|representations]] of a [[group (mathematics)|group]] ''G'' on [[finite-dimensional]] [[vector space]]s over a given [[field (mathematics)|field]] ==(''E'', ''G'')-rings and the functor ''D''== ...

{{Short description|The probability that two uniform random elements of a finite group commute with each other}} ...[[Algebraic structure|algebraic structures]] such as [[Ring (mathematics)|rings]].<ref name=":1" /> ...

...in [[Module (mathematics)|modules]] over semiprimary [[ring (mathematics)|rings]]. A ring ''R'' (with&nbsp;1) is called '''semiprimary''' if ''R''/''J''(' ...[[noetherian ring|Noetherian]]. The analogous statement for left Artinian rings holds as well. This is not true in general for Artinian modules, because th ...

...' is a [[faithful module]], ''R'' and ''R''_''M'' are isomorphic rings. '''Theorem''': If ''A'' is a finite-dimensional central simple algebra over a field ''F'' and ''B'' is a simple ...

==Statement for rings== ==Statement for group rings== ...

...th>n</math> = 0. Another example of a finite-type homomorphism that is not finite is <math>\mathbb{C}[t] \to \mathbb{C}[t][x,y]/(y^2 - x^3 - t)</math>. ...rt needed|Mathematics |talk=distinction between finite type and locally of finite type|date=August 2023}} ...

...g ''R''. Consider their [[tensor algebra]], the direct sum of all possible finite tensor products of ''A'', ''B''; explicitly, <math>T = \bigoplus_{n=0}^{\in A finite free product is defined similarly.<!-- no idea for infinite free product. I ...

{{about|perfect rings as introduced by Hyman Bass|perfect rings of characteristic p generalizing perfect fields|perfect field}} ...there exist rings which are perfect on one side but not the other. Perfect rings were introduced in [[Hyman Bass|Bass]]'s book.{{sfn|Bass|1960}} ...

== Difference rings, difference fields and difference algebras == ...lgebra structure such that <math>K\to R</math> is a morphism of difference rings, i.e. <math>\sigma\colon R\to R</math> extends <math>\sigma\colon K\to K</m ...

...as only been [[mathematical proof|proven]] for special types of Noetherian rings, so far. Examples exist to show that the [[conjecture]] can fail when the r | title = Structure of rings ...

{{short description|Abstract ring with finite number of elements}} ...algebra]], a '''finite ring''' is a [[ring (mathematics)|ring]] that has a finite number of elements. ...

{{short description|About extensions of one-dimensional Noetherian rings (commutative algebra)}} ...of ''K''.<ref>If <math>A\subset B</math> are rings, we say that ''B'' is a finite extension of ''A'' if ''B'' is a finitely generated ''A'' module.</ref> If ...

...</math> of [[scheme (mathematics)|schemes]] such that (a) it is locally of finite presentation and (b) for each <math>x \in X</math> and <math>y = f(x)</math ...}_Q)</math> and <math>(\mathcal{O}_P, \mathfrak{m}_P)</math> are the local rings at ''Q'' and ''P'' of ''Y'' and ''X''. Since <math>\mathcal{O}_P</math> is ...

{{Short description|Type of finite commutative rings}} ...ucted from the ring <math>\mathbb{Z}/p^n\mathbb{Z}</math> similar to how a finite field <math>\mathbb{F}_{p^r}</math> is constructed from <math>\mathbb{F}_p< ...