29-11-2012, 12:17 PM
RINGS
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Introduction
Groups, rings and fields are building blocks of modern algebra.
They are algebraic structures used to generalize the main properties of integers, real numbers, complex numbers and square matrices.
Difference between a group and a ring is that a group is a non empty set with single binary operator * where as a ring is a nonempty set with 2 different binary operator + and *.
DEFINITION:
A ring is an abelian group R with binary operation + (“addition”), together with a second binary operation ・ (“multiplication”). The operations satisfy the following axioms:
1. Multiplication is associative
2. The Distributive Law holds
Integral domain
An integral domain is a commutative ring with identity in which the product of any two nonzero elements is not equal to zero.
A ring whose non-zero elements form an abelian group under multiplication, is called a field. So every field is an integral domain and every integral domain is a commutative ring.
DIVISION RING : R is a division ring or a skew field if every non-zero element of R has a multiplicative inverse in R.
Subring
A subring of a ring is a subset which is closed under the operation of addition and multiplication and contains the multiplicative identity.
A subring of a ring (R, +, *) is a subgroup of (R, +) which contains the mutiplicative identity and is closed under multiplication.
CHARECTERISTICS OF A RING
The characteristic of a ring R, often denoted char®, is defined to be the smallest number of times one must use the ring's multiplicative identity element (1) in a sum to get the additive identity element (0); the ring is said to have characteristic zero if this repeated sum never reaches the additive identity.
That is, char® is the smallest positive number n such that n.a=0 for all a ∈R then the least such positive integer is char of the ring
BASIC PROPERTIES OF POLYNOMIAL RINGS
Let R be a ring. The polynomial ring in one variable, is the set of all formal polynomials in one variable, with addition and multiplication defined as usual.
We are in particular interested in the polynomial ring in one variable over a field.
If the product of two polynomials is the zero polynomial, then one of the polynomials must be the zero polynomial.