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 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).
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 x_{1}, ... ,x_{n}
for some n such that every element in J has the form
a_{1}x_{1} + ... + a_{n}x_{n}
for some elements a_{1}, ... ,a_{n} of A.
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 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.
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.
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.
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 }.
An integral domain A in which every nonzero
noninvertible element can be factorized
into a product of irreducible elements
p_{1}p_{2} ... p_{n}
in an essentially unique way, i.e. so that the collection of ideals
{ p_{1}A, ... , p_{n}A } is unique.
Related pages in this website
Sets - how to construct sets of integers, reals,
etc.