Glossary of ring theory
Please refer to ring theory for a general description of the subject.A ring is an abelian group (R,+) together with a distributive operation *. + is referred as the addition and * is referred as the multiplication. A ring is unital if it has a multiplicative identity.
Throughout this article, all rings are unital and that the additive inverse 0 is different from the multiplicative identity 1.
Basic definitions
Unit. An element r of R is a unit if there exists an element r-1 such that rr-1=r-1r=1. r-1 is unique and is called the multiplicative inverse of r.
The set of units forms a group under multiplication.
Zero divisor. An nonzero element r of R is said to be a zero divisor if there exists s &ne 0 such that sr=0 or rs=0.
Torsion. The set of zero divisor in R. A ring without zero divisor is torsion-free.
Subring. A subset S of (R,*,+) which remains a ring while + and * are restricted on S is called a subring of R.
Given a subset T of R, we denoted by <T> the smallest subring of R containing T.
Ideal. A left ideal I of R is a subring such that aI⊂ I for all a∈R. A righ ideal is those subring that Ia⊂I. An ideal is a subring which is both a left ideal and a right ideal.
Ring homomorphism. These are functions f: (R,+,*) → (S,⊕,×) that have the special property that
- f(a + b) = f(a) ⊕ f(b)
- f(a * b) = f(a) × f(b)
Kernel of a ring homomorphism. It is the preimage of the multiplicative identity in the codomain of a group homomorphism. Every ideal is the kernel of a ring isomorphism and vice versa.
Ring isomorphism[?]. Ring homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism.
Isomorphic rings. Two rings are isomorphic if there exists a ring isomorphism mapping from one to the other. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.
Factor ring. Given a ring R and a normal subgroup I of R, the factor ring is the set R/I of left cosets {aI : a∈I'} together with operations aI+bI=(a+b)I and aI*bI=abI. The relationship between ideals, homomorphisms, and factor rings is summed up in the fundamental theorem on homomorphisms.
Direct product and direct sums[?]. They are ways to combining subrings, please refer to the corresponding links for explanation.
Types of rings
Commutative ring. A ring R is commutative if the multiplication is commutative, i.e. gh=hg for all g,h∈R.
Integral domain. It is a commutative ring without zero divisor.
Division ring or skew field. It is a ring of which every nonzero element is a unit.
Noetherian ring. Rings satisfying ascending chain condition for ideals.
Artinian ring[?]. Rings satisfying descending chain condition for ideals.
Dedekind domain. It is an integral domain of which every ideal is finitely generated.
Field. A commutative division ring. Every finite division ring is a field. Field theory[?] is indeed an older mathematics branch than ring theory.
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