defs 1+2 Flashcards
a ring
A ring is a set R and two binary operations, written + and ×, on R which satisfies the following conditions:
(R1) < R, + > is an abelian group with identity 0
(R2) × is associative.
(R3) × is distributive over +:
ie. a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a) for all a, b, c ∈ R.
(R4) there exists an element 1 ∈ R, different from 0, that is an identity for ×.
abelian
commutative, order doesn’t matter
subring
Let R be a ring and S ⊆ R. Then S is a subring of R if it is
a ring in its own right with respect to the same addition and multiplication as in R and S contains 1R
subring test
Let R be a ring and S ⊆ R. Then S is a subring of R, iff:
(i) 1R ∈ S;
(ii) r + s, r × s ∈ S, for all r, s ∈ S;
(iii) −r ∈ S for all r ∈ S
polynomial ring
Let R be a ring. The ring of polynomials R[X] in the indeterminate X is
defined as follows:
Elements: Formal linear combinations of the form
SUM [i≥0] aiX^i
where the coefficients ai ∈ R for i = 0, 1, … and only finitely many of these coefficients are non-zero. When writing specific polynomials we normally don’t include the terms with zero coefficients.
Equality:
SUM [i≥0] aiX^i = SUM [i≥0] biX^i ⇔ ai = bi for all i ≥ 0.
Addition:
SUM [i≥0] aiX^i + SUM [i≥0] biX^i =SUM [i≥0] (ai+bi) X^i
Multiplication:
[SUM [i≥0] aiX^i ] [SUM [i≥0] biX^i] =
SUM [k≥0] {SUM [i+j=k] aibj} X^k
The zero element is SUM [i≥0] 0X^i= 0
The one is 1X^0 +SUM [i≥0] 0X^i = 1
degree of polynomial
deg(f) = the largest i such that ai =! 0 and we let deg(f) = −∞ if f = 0.
cartesian product
If R1 and R2 are rings then the Cartesian product R1×R2
with operations + and × defined by
(r1, r2) + (s1, s2) = (r1 + s1, r2 + s2) and
(r1, r2) × (s1, s2) = (r1 × s1, r2 × s2) is a ring. The element (0, 0) is the zero of this ring and the element (1, 1) is the one of R1 × R2.
n · a
n · a = a + a + … + a (n times)
0 · a = 0 and setting n · a = (−n) · (−a) if n < 0. From the definition we get: (n + m) · a = n · a + m · a n · (a + b) = n · a + n · b n · (m · a) = (nm) · a where n, m ∈ Z and a, b ∈ R.
characteristic
The characteristic, char(R), of a ring R is the least positive integer n such that 1 + · · · + 1 = 0 (n 1s), that is, such that n · 1 = 0.
If there is no such n, then the characteristic of R is defined to be 0.
For example, Z has characteristic 0, as have the rings Q and C. On the otherhand char(Zn) = n
zero divisor
A non-zero element r ∈ R is a zero-divisor if there is a
non-zero element s ∈ R with rs = 0 or sr = 0.
domain
The ring R is a domain if, for all r, s ∈ R
rs = 0 ⇒ r = 0 or s = 0.
So a domain is a ring with no zero-divisors
A commutative domain is called an integral domain. (Recall that a ring R is said to be commutative if the multiplication is commutative: rs = sr for all r, s ∈ R.)
division ring
A division ring is a ring in which every non-zero element has a right inverse and a left inverse, ie:
for every r ∈ R there is s ∈ R such that rs = 1 (r is right invertible)
and there is t ∈ R such that tr = 1 (r is left invertible).
In this case (see below) s = t and we write r^-1 for this inverse of r and say just that r is invertible or that r is a unit.
field
A field is a commutative division ring
nilpotent
An element r of a ring R is nilpotent if there is some integer n >= 1 with r^n = 0
The least such n is the index of nilpotence of r
idempotent
An element r ∈ R is idempotent if r^2 = r. For example 0 and 1 are idempotent in any ring
