In mathematics, a subring is a subset of a ring, which contains the multiplicative identity and is itself a ring under the same binary operations. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity (if it exists). This leads to the added advantage that ideals become subrings (see below). In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

A subring of a ring (R, +, *) is a subgroup of (R, +) which contains the mutiplicative identity and is closed under multiplication. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X]. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

The ring Z has no subrings (with multiplicative identity) other than itself.

Every ring has a unique smallest subring, isomorphic to either the integers Z or some ring Z/nZ with n a nonnegative integer (see characteristic). In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

The subring test states that for any ring, a nonempty subset of that ring is itself a ring if it is closed under multiplication and subtraction, and has a multiplicative identity. In Abstract algebra, the subring test is a Theorem that states that for any ring, a nonempty Subset of that ring is a Subring if it

Subring generated by a set

Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T. ) S is said to be the subring of R generated by X. In Mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings If S = R, we may say that the ring R is generated by X.

Relation to ideals

Proper ideals are never subrings since if they contain the identity then they must be the entire ring. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. For example, ideals in Z are of the form nZ where n is any integer. These are subrings if and only if n = ±1 (otherwise they do not contain 1) in which case they are all of Z.

If one omits the requirement that rings have a unit element, then subrings need only contain 0 and be closed under addition, subtraction and multiplication, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

• The ideal I = {(z,0)|z in Z} of the ring Z × Z = {(x,y)|x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z have no multiplicative identity.

Profile by commutative subrings

A ring may be profiled by the variety of commutative subrings that it hosts:

• The quaternion ring H contains only the complex plane as a planar subring
• The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, as well as the ordinary complex plane
• The ring of 3 x 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). In Mathematics, commutativity is the ability to change the order of something without changing the end result Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the A variety of dualities in mathematics are listed at Duality (mathematics. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 x 3 matrices. In Mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

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