Question

If A,B belongs to a and a is an algebra of subset of X prove that A-B belongs to a​

Answer 1

Answer:

If a, b are two ring elements with a, b ≠ 0 but ab = 0 then a and b are called zero-divisors.

Example

In the ring Z6 we have 2.3 = 0 and so 2 and 3 are zero-divisors.

More generally, if n is not prime then Zn contains zero-divisors.

Definition

An integral domain is a commutative ring with an identity (1 ≠ 0) with no zero-divisors.

That is ab = 0 ⇒ a = 0 or b = 0.

Examples

The ring Z is an integral domain. (This explains the name.)

The polynomial rings Z[x] and R[x] are integral domains.

(Look at the degree of a polynomial to see how to prove this.)

The ring {a + b√2 | a, b ∈ Z} is an integral domain.

(Proof?)

If p is prime, the ring Zp is an integral domain.

(Proof?)

Step-by-step explanation:

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