intersection of subrings of R is itself a subring of R; therefore, the subring generated by X (denoted here as S) is indeed a subring of R. This subring S is the...

Southern University and A&M College (Southern University, Southern, SUBR or SU) is a public historically black land-grant university in Baton Rouge, Louisiana...

intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, and...

Conversely, if R is an associative algebra over a commutative subring S, then S is a subring of the center of R, and if S happens to be the center of R,...

{\displaystyle x} . If A is a subring of a ring B, and b is an element of B, then A[b] denotes the subring of B generated by A and b. This subring consists of all the...

to R. The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb...

{\displaystyle B.} The integral closure of any subring A {\displaystyle A} in B {\displaystyle B} is, itself, a subring of B {\displaystyle B} and contains A ...

tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed (see below)...

In algebra, the fixed-point subring R f {\displaystyle R^{f}} of an automorphism f of a ring R is the subring of the fixed points of f, that is, R f =...

A} is a subring of R {\displaystyle R} such that I ⊆ A ⊆ R {\displaystyle I\subseteq A\subseteq R} , then A / I {\displaystyle A/I} is a subring of R /...

characteristic zero contains a subring isomorphic to ⁠ Z {\displaystyle \mathbb {Z} } ⁠, which is its smallest subring. Z {\displaystyle \mathbb {Z} }...

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly...

NA(S) is the largest Lie subring of A in which S {\displaystyle S} is a Lie ideal. If S {\displaystyle S} is a Lie subring of a Lie ring A, then S ⊆...

one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These...

a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex multiplication...

Artinian, Noetherian, prime. If S is a subring of R, then Mn(S) is a subring of Mn(R). For example, Mn(Z) is a subring of Mn(Q). The matrix ring Mn(R) is...

In mathematics, an order in the sense of ring theory is a subring O {\displaystyle {\mathcal {O}}} of a ring A {\displaystyle A} , such that A {\displaystyle...

Another natural question is: "When is a subring of a division ring right Ore?" One characterization is that a subring R of a division ring D is a right Ore...

fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F such that either x or x−1 belongs to D for every nonzero x in F, then...

elements x of R such that xr = rx for all r in R. The center is a commutative subring of R. The center of a Lie algebra L consists of all those elements x in...

ring GR(pn, r) contains a unique subring isomorphic to GR(pn, s) for every s which divides r. These are the only subrings of GR(pn, r). The units of a Galois...

closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by...

ring is an algebra over its differential subring. This is the natural structure of an algebra over its subring. Ring ( Q { y , z } , ∂ y ) {\textstyle...

of several similar results. These results concern the centralizer of a subring S of a ring R, denoted CR(S) in this article. It is always the case that...

homomorphism R → S exists. If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f :...

exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they...

are defined similarly. 2.  A maximal subring is a subring that is maximal among proper subrings. A "minimal subring" can be defined analogously; it is unique...

ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group; however there are rings that...

ring A is a subring of a commutative Noetherian ring B such that B is faithfully flat over A (or more generally exhibits A as a pure subring), then A is...

addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted...