Question 4 Flashcards
(12 cards)
Definition: Principal ideal domain
A principal ideal domain (PID) is an integral domain in which
every ideal is principal.
Definition: Associates
We say x,y∈R are associates if x= uy for some
unit u∈R×
Definition : Irreducible
An element p∈R is irreducible if p is non-zero,
not a unit, and not the product of two non-units.
Definition: Unique factorisation domain
An integral domain R is a unique
factorisation domain (UFD) if
(1) every non-zero non-unit can be written as a product of irreducible elements,
(2) if p1···pm = q1···qn with p1,…,pm,q1,…,qn irreducible then m = n and
the lists can be reordered so that pi and qi are associates for each i= 1,…,n.
It is not true that all integral domains are unique factorisation domains
Definition: Prime
An element p∈R is prime if p is non-zero and (p) is a
prime ideal; equivalently, p is prime if p is a non-zero non-unit and p|xy implies
p|x or p|y (for all x,y∈R).
Let R be a principal ideal domain. Then R is a unique factorisation
domain.
Fundamental theorem of arithmetic
Z(bar) is a UFD.
Definition : degree of polynomial
The degree deg(f) of f is the largest index n such that a_n <>0, assuming f<> 0. The degree of the zero polynomial is defined to be −∞.
(2) A term of f is a_iX_i with a_i <> 0.
(3) If f <> 0, the leading term of f is anXn where n = deg(f). The leading
coefficient of f is a_n.
(4) A non-zero polynomial is monic if its leading coefficient is 1.
(5) The constant term of f is a_0.
Definition : root of a polynomial
A root of a polynomial f ∈R[X] is an element x∈R such that
f(x) = 0.
Definition : nilpotent
An element of r a ring R is said to be nilpotent if rn = 0 for
some n∈N. The smallest such n is called the index.
McCoy’s Theorem
For a commutative ring R, a polynomial
f ∈R[X] is a zero divisor if and only if there is some non-zero r ∈R such that
rf = 0.
