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.
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.
A Division Algebra is a nontrivial ring (not necessarily commutative) in which
all nonzero elements are invertible.
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.
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).
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.
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.
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.
A Gaussian integer is a complex number whose real and imaginary parts are integers.
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
- f(0) = 0,
- f(1) = 1,
- f(a+a') = f(a) + f(a'),
- f(aa') = f(a)f(a').
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.
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.
An integral domain is a nontrivial commutative ring
in which the product of nonzero elements is nonzero.
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.
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.
An ideal is maximal if it is proper,
but not contained in any other proper ideal.
A monic polynomial is one whose term of highest degree has 1 as coefficient.
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.
A polynomial expression is one constructed using +, - ,o,1 and multiplication.
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.
A polynomial f(t) in A[t] determines a function A --> A given by a |--> f(a).
The formal power series
a0 + a1t + . . . + antn + . . .
with coefficients from a ring A in an indeterminate t consititute
a ring A[[t]].
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.
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.
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.
An integral domain in which all ideals are
principal.
An ideal is proper if it is not the whole 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
- 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
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.
A ring is trivial if it has only one element ( so 0 = 1 in such a ring).
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.