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Algebraic structure → Ring theory Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring
• Terminal ring
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring
• Integers modulo pn
• Prüfer p-ring
• Base-p circle ring
• Base-p integers
• p-adic rationals
• Base-p real numbers
• p-adic integers
• p-adic numbers
• p-adic solenoid
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
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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 that shares the same multiplicative identity as R. (Note that a subset of a ring R need not be a ring.) 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.
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...
Southern University and A&M College (Southern University, Southern, SUBR or SU) is a public historically black land-grant university in Baton Rouge, Louisiana...
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)...
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 mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly...
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...
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...
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...
{\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 ...
{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...
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...
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...
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...
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 :...
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...
{\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...
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...
addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted...
ring of a field K is a subring R such that for every non-zero element x of K, at least one of x and x−1 is in R. Any such subring will be a local ring....