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Ring Theory


For any ring A there is a unique homomorphism Z —> A, where Z denotes the ring of integers. If the kernel of this homomorphism is nZ we say that A has characteristic n.

Content of a Polynomial

The content of a polynomial over a UFD is the highest common factor of its coefficients.


Two elements x,y of a ring A are coprime if xA+yA = A. In other words, if there exist elements a,b of A such that ax+by = 1.

Division Algebra

A Division Algebra is a nontrivial ring (not necessarily commutative) in which all nonzero elements are invertible.

Euclidean Algorithm

The Euclidean algorithm is a method for finding the highest common factor d of two elements x,y in a Euclidean Domain. If y=0 then d is x. If x=0 then d is y. Otherwise, keep subtracting the smaller from the larger until one element is zero.

Euclidean Domain

A Euclidean Domain is an integral domain with a nonnegative integer valued function d defined on nonzero elements such that for any two elements x,y with y nonzero, there exist elements q,r such that x = yq+r and either r = 0 or d(r) < d(y).

Fermat's Theorem

An odd prime number is a sum of two squares if and only if it is congruent to 1 modulo 4.


A field is a nontrivial commutative ring in which every nonzero element is invertible.

Field of Fractions

Given an integral domain A there exists a field Q(A) containing A as a subring with the property that for every x in Q(A) there are elements a,b of A so that xb=a. Q(A) is unique up to isomorphism. It is the field of fractions of A.

Finitely Generated Ideal

An ideal J in a commutative ring A is finitely generated if it contains elements x1, ... ,xn for some n such that every element in J has the form a1x1 + ... + anxn for some elements a1, ... ,an of A.

Gaussian Integer

A Gaussian integer is a complex number whose real and imaginary parts are integers.

Highest Common Factor

A highest common factor of two elements x,y in a ring A is an element d which is divisible by all those elements of A which divide both x and y. It may not exist.


A homomorphism from a ring A to a ring B is a function f: A —> B such that


An ideal J in a ring A is a subset of A such that


The image of a homomorphism f: A —> B is the subring of B whose elements have the form f(a), for some a in A.

Integral Domain

An integral domain is a nontrivial commutative ring in which the product of nonzero elements is nonzero.

Invertible Element

An element x in a ring A is invertible if there exists an element y in A such that xy = yx = 1. Then y is unique and we write it as x-1.

Irreducible Element

An element x in an integral domain A is irreducible if it is not invertible and if x = yz implies that either y or z is invertible.


An isomorphism of rings is a bijective homomorphism.


The kernel of a homomorphism of rings f: A —> B is the ideal in A consisting of those elements a in A such that f(a) = 0.

Maximal Ideal

An ideal is maximal if it is proper, but not contained in any other proper ideal.

Monic Polynomial

A monic polynomial is one whose term of highest degree has 1 as coefficient.

Noetherian Ring

A ring is Noetherian if it has no infinite strictly increasing chain of ideals. This is equivalent to the condition that all ideals are finitely generated.

Polynomial Expression

A polynomial expression is one constructed using +, - ,o,1 and multiplication.

Polynomial Ring

The polynomial ring A[t] over a ring A consists of all the formal polynomials with coefficients from A in an indeterminate symbol t. It has the universal property that homomorphisms A[t] —> B are in bijective correspondence with pairs (f,b) where f: A —> B is a homomorphism and b is an element of B.

Polynomial Function

A polynomial f(t) in A[t] determines a function A —> A given by a |—> f(a).

Power Series

The formal power series a0 + a1t + ... + antn + ... with coefficients from a ring A in an indeterminate t consititute a ring A[[t]].

Prime Element

An element x in an integral domain is prime if, for any product yz in the ring, if x divides yz then either x divides y or x divides z.

Prime Element

An ideal J in a ring A is prime if the elements of A not in J are closed under multiplication. In other words, if xy in J implies that either x is in J or y is in J.

Principal Ideal

The Principal ideal generated by an element x in a commutative ring A is the set xA = { xa | a in A } of multiples of x.

Principal Ideal Domain (PID)

An integral domain in which all ideals are Principal.

Proper ideal

An ideal is proper if it is not the whole ring.

Quotient Ring

An ideal J of a ring A gives rise to a surjective homomorphism A —> A/J taking an element a of A to the element (a+J) of A/J. The expression a+J denotes the coset { a+x | x in J }.


A ring is a set A with binary operations x+y (addition), xy (multiplication), a unary operation -x (negation) and constants 0, 1 such that A ring is commutative if it also satisfies


A subset B of a ring A is a subring if

Trivial Ring

A ring is trivial if it has only one element ( so 0 = 1 in such a ring).

Unique Factorization Domain (UFD)

An integral domain A in which every nonzero noninvertible element can be factorized into a product of irreducible elements p1p2 ... pn in an essentially unique way,
i.e. so that the collection of ideals { p1A, ... , pnA } is unique.

Related pages in this website

Sets - how to construct sets of integers, reals, etc.

Group - a set closed under one operation.

Fermat's Theorems -- Fermat's Little Theorem, in particular.


The webmaster and author of this Math Help site is Graeme McRae.