subset of R, the intersection of all subrings of R containing X is a subring S of R. This subring is the smallest subring of R containing X. ("Smallest" means...

{\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...

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...

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

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...

R be a ring. Let S be a subring of R, and let I be an ideal of R. Then: The sum S + I = {s + i | s ∈ S, i ∈ I } is a subring of R, The intersection S ∩ I...

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 =...

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)...

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

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...

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

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...

{Z} } is not a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals...

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

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...

{\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 ...

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...

integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain....

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...

algebraic integers, and, more generally of integral elements. Let R be a subring of a field F; this implies that R is an integral domain. An element a of...

integers, which contains the ring of rationals with odd denominators as a subring. When using the "shortcut" definition of the Collatz map, it is known that...

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...

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 :...

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

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...

{\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...

has the following properties. It is an integral domain, since it is a subring of a field, or since the first term of the series representation of the...

analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as OK. These are...