# Ring Theory

### Characteristic

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.

### Coprime

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.

### Field

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.

### Homomorphism

A homomorphism from a ring A to a ring B is a function f: A —> B such that
• f(0) = 0,
• f(1) = 1,
• f(a+a') = f(a) + f(a'),
• f(aa') = f(a)f(a').

### Ideal

An ideal J in a ring A is a subset of A such that
• 0 is in J,
• if x,y are in J then x+y is in J,
• if x is in J then ax and xa are in J for any a in A.

### Image

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.

### Isomorphism

An isomorphism of rings is a bijective homomorphism.

### Kernel

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 }.

### Ring

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
• x + (y + z) = (x + y) + z
• x(yz) = (xy)z
• 0 + x = x + 0 = x
• 1x = x1 = x
• x + y = y + x
• x + (-x) = 0
• x(y + z) = (xy) + (xz)
• (y + z)x = (yx) + (zx)
A ring is commutative if it also satisfies
• xy = yx

### Subring

A subset B of a ring A is a subring if
• 0 and 1 are in B,
• B is closed under addition and subtraction,
• B is closed under multiplication.

### 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.