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,...

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 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...

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 =...

{\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 ...

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} }...

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 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 ⊆...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 :...

{\displaystyle n>0} , then this ring is always a subring of R {\displaystyle \mathbb {R} } , otherwise, it is a subring of C . {\displaystyle \mathbb {C} .} The...

of the polynomial B. The obvious choice for such a subring is the centralizer Z of A, the subring of all matrices that commute with A; by definition A...

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

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...

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