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In mathematics, a **subring** of *R* is a subset of a ring that is itself a ring when binary operations of addition and multiplication on *R* are restricted to the subset, and which shares the same multiplicative identity as *R*. For those who define rings without requiring the existence of a multiplicative identity, a subring of *R* is just a subset of *R* that is a ring for the operations of *R* (this does imply it contains the additive identity of *R*). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of *R*). With definition requiring a multiplicative identity (which is used in this article), the only ideal of *R* that is a subring of *R* is *R* itself.

A subring of a ring (*R*, +, ∗, 0, 1) is a subset *S* of *R* that preserves the structure of the ring, i.e. a ring (*S*, +, ∗, 0, 1) with *S* ⊆ *R*. Equivalently, it is both a subgroup of (*R*, +, 0) and a submonoid of (*R*, ∗, 1).

The ring and its quotients have no subrings (with multiplicative identity) other than the full ring.

Every ring has a unique smallest subring, isomorphic to some ring with *n* a nonnegative integer (see characteristic). The integers correspond to *n* = 0 in this statement, since is isomorphic to .

The **subring test** is a theorem that states that for any ring *R*, a subset *S* of *R* is a subring if and only if it is closed under multiplication and subtraction, and contains the multiplicative identity of *R*.

As an 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*].

If *S* is a subring of a ring *R*, then equivalently *R* is said to be a **ring extension** of *S*, written as *R*/*S* in similar notation to that for field extensions.

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*. If *S* = *R,* we may say that the ring *R* is *generated* by *X*.

Proper ideals are subrings (without unity) that are closed under both left and right multiplication by elements of *R*.

If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, 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.

If *I* is a prime ideal of a commutative ring *R*, then the intersection of *I* with any subring *S* of *R* remains prime in *S*. In this case one says that *I***lies over***I* ∩ *S*. The situation is more complicated when *R* is not commutative.

A ring may be profiled^{[ clarification needed ]} 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 × 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). 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 × 3 matrices.

In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, specifically abstract algebra, an **integral domain** is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element *a* has the cancellation property, that is, if *a* ≠ 0, an equality *ab* = *ac* implies *b* = *c*.

In algebra, a **prime ideal** is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

In ring theory, a branch of abstract algebra, an **ideal** of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

In mathematics, **rings** are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a *ring* is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, more specifically ring theory, the **Jacobson radical** of a ring is the ideal consisting of those elements in that annihilate all simple right -modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by or ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in.

In ring theory, a branch of abstract algebra, a **commutative ring** is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In mathematics, specifically ring theory, a **principal ideal** is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in

In abstract algebra, more specifically ring theory, **local rings** are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. **Local algebra** is the branch of commutative algebra that studies commutative local rings and their modules.

In mathematics, an **algebra over a field** is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In the branch of abstract algebra known as ring theory, a **unit** of a ring is any element that has a multiplicative inverse in : an element such that

In abstract algebra, a **matrix ring** is a set of matrices with entries in a ring *R* that form a ring under matrix addition and matrix multiplication. The set of all *n* × *n* matrices with entries in *R* is a matrix ring denoted M_{n}(*R*). Some sets of infinite matrices form **infinite matrix rings**. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs.

In mathematics, **complex multiplication** (**CM**) is the theory of elliptic curves *E* that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties *A* having *enough* endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

In ring theory, a ring R is called a **reduced ring** if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, *x*^{2} = 0 implies *x* = 0. A commutative algebra over a commutative ring is called a **reduced algebra** if its underlying ring is reduced.

In abstract algebra, a **valuation ring** is an integral domain *D* such that for every element *x* of its field of fractions *F*, at least one of *x* or *x*^{ −1} belongs to *D*.

In algebra, the **zero-product property** states that the product of two nonzero elements is nonzero. In other words, it is the following assertion:

If , then or .

In ring theory, a branch of mathematics, **semiprime ideals** and **semiprime rings** are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called **radical ideals**.

In commutative algebra, an element *b* of a commutative ring *B* is said to be **integral over***A*, a subring of *B*, if there are *n* ≥ 1 and *a*_{j} in *A* such that

This is a **glossary of commutative algebra**.

- Iain T. Adamson (1972).
*Elementary rings and modules*. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3. - Page 84 of Lang, Serge (1993),
*Algebra*(Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001 - David Sharpe (1987).
*Rings and factorization*. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.

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