Factorisation Of Rings Flashcards
What is a ring?
A ring R is an abelian group under addition that is associative with respect to multiplication —
. a(bc) = (ab)c for a, b, c, € R
— and is left-and-right distributive over addition:
. a(b+c) = ab + ac,
. (b+c)a = ba + ca
What is a zero divisor?
Let R be a ring, and a, b € R; if ab=0, then a and b are said to be zero divisors — left and right zero divisors respectively.
What is a proper zero divisor?
If ab = 0, and a ≠ 0, b ≠ 0, then a and b are proper zero divisors.
List rings that have no proper zero divisors.
They are:
Z, Q, C, and R, which are the set of integers, rational numbers, complex numbers, and real numbers respectively.
What is an integral domain?
A ring R is said to be an integral domain if:
1. It is commutative under multiplication,
2. It possesses no proper zero divisors.
Prove that if R is a commutative ring with identity (1) and an ideal A contains a unit, then the R = A.
As R is a ring containing a unit, u, u ∈ R: uu* = 1 for some u* ∈ R.
Suppose A is an ideal containing u. Then,
1 = uu* ∈ AR ⊆ A by definition of ideals, implying 1 ∈ A.
Every element of R, r ∈ R, can then be expressed as:
r = (1)r ∈ AR ⊆ A
Thus if an ideal contains a unit, then the ideal contains/equals the entire ring.
What is a field?
A field is a commutative ring where every non-zero element is a unit.
What is an ideal?
A (two-sided) ideal A of a ring R is a subring where for every a ∈ A and r ∈ R, ar and ra are contained in A.
What is a unit?
Let R be a commutative ring with the identity element 1. If for an element u ∈ R there exists u* such that,
uu* = 1, then u is said to be a unit.
Alternatively, a unit is a u ∈ R: u|1
Define the divisibility of a commutative ring.
R is a commutative ring, with a, b ∈ R, a is said to divide b if there exists c ∈ R:
b = ac
What is an irreducible?
Let R be an integral domain, and P ∈ R, then P is irreducible if, for a, b ∈ R:
1. P ≠ 0, P ∉ U(R) (not an element of the set of units in R)
2. P = ab, where either a or b are units.
