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 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[tex]\neq0[/tex] and v [tex]\neq0[/tex] 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 [tex]\neq0[/tex]such that au=1By definition of unit b is a zero divisor then there exist an element [tex]v\neq0[/tex]such that bv=0 where [tex]b\neq0[/tex]By definition of zero divisor(ab)(uv)=b(au)v    ( because ring is commutative)(ab)(uv)=b.1.v=bv=0Hence, ab is a zero divisor.If a is unit and b is a zero divisor then ab is a zero divisor.