Page 51 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

also Notes
a b a b a a a b b c
A B 1 1 2 2 1 2 1 2 2 1 M
0 c 0 c 0 c c
1 2 1 2
M is subring of R.

Self Assessment

9. Show that the set of even integers including zero is a commutative ring with zero-divisors
under the usual addition and multiplication.

10. Prove that the ring R = {0, 1, 2, 3, 4, 5, 6, 7} under the addition and multiplication modulo
8 is a commutative ring without zero divisors.
11. Prove that set I of integers is a subring of R, the set of real numbers.

2
2
12. If a, b belong to a ring R and (a + b) = a + 2ab + b , then show that R is a commutative ring.
2
Ideals
Definition: Let (R, +,.) be any ring and S a subring of R, then S is said to be right ideal of R if
a S, b R a b S and left ideal of R if a S, b R b a S.
Thus a non-empty subset S or R is said to be a ideal of R if:

(i) S is a subgroup of R under addition.
(ii) a S and b R, both a b and ba S.
Principal Ideals: If R is a commutative ring with unity and a R, the ideal {ax : x R} is called the
principal ideal generated by a and is denoted by (a), thus (a) stands for the ideal generated by a.
Principal Ideal Ring: A commutative ring with unity for which every ideal is a principal ideal
is said to be a principal ideal ring.

Prime Ideal: Let R be a commutative ring. An ideal P of ring R is said to be a prime ideal of R if
ab P , ,b R a P or b P .
a

Example 25: In the commutative ring of integers I, the ideal P = {5r : r I} is a prime ideal
since if ab P , then 5 | ab and consequently 5 | a or 5 | b as 5 is prime.

Integral Domain

Definition: A commutative ring with unity is said to be an integral domain if it has no zero-
divisors. Alternatively a commutative ring R with unity is called an integral domain if for all a,
b R, a b = 0 a = 0 or b = 0.

Examples:
(i) The set I of integers under usual addition and multiplication is an integral domain as for
any two integers , ;a b ab 0 a 0 or b 0.

(ii) Consider a ring R = {0, 1, 2, 3, 4, 5, 6, 7} under the addition and multiplication modulo 8.
This ring is commutative but it is not integral domain because 2 R , 4 R are two
non-zero elements such that 2.4 = 0 (mod 8).

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