Question

Let R be a communtative ring and a, b elements in R. Prove that if a and b are units, then so is ab. What can we say about ab when a is a unit and b is a zero divisor? Prove your claim.

Answer 1

Answer with explanation:

Let R be a communtative ring .

a and b elements in R.Let a and b are units

1.To prove that ab is also unit in R.

Proof: a and b are units.Therefore,there exist elements u
`\\eq0` and v
`\\eq0` such that

au=1 and bv=1 ( by definition of unit )

Where u and v are inverse element of a and b.

(ab)(uv)=(ba)(uv)=b(au)(v)=bv=1 ( because ring is commutative)

Because bv=1 and au=1

Hence, uv is an inverse element of ab.Therefore, ab is a unit .

Hence, proved.

2. Let a is a unit and b is a zero divisor .

a is a unit then there exist an element u
`\\eq0`

such that au=1

By definition of unit

b is a zero divisor then there exist an element
`v\\eq0`

such that bv=0 where
`b\\eq0`

By definition of zero divisor

(ab)(uv)=b(au)v ( because ring is commutative)

(ab)(uv)=b.1.v=bv=0

Hence, ab is a zero divisor.

If a is unit and b is a zero divisor then ab is a zero divisor.