Discrete Mathematical Structures by Pearson

7-46 Discrete Mathematical Structures

Note 7.10 The sum of two subrings may not be a subring. For example, the multiples of 2 and

multiples of 3 are subrings of the ring of integers. The sum will contain 2 and 3, but their sum

2 + 3 = 5 is not in the sum. Hence, the sum is not a subring.

7.12.1 Ideal

Let S be a non-empty subset of a ring R such that S is an additive subgroup of R, that is, a - b ∈ S,

∀a, b ∈ S, then

1. S is a left ideal of R iff ra ∈ S for each r ∈ R and a ∈ S.

2. S is a right ideal of R iff ar ∈ S for each r ∈ R and a ∈ S.

3. S is an ideal of R iff it is both a left ideal and a right ideal of R.

If the ring is commutative left and right ideals coincide.

Example 1 The subring of even integers is an ideal of the ring of integers. ...