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=1
By 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=0
Hence, ab is a zero divisor.
If a is unit and b is a zero divisor then ab is a zero divisor.
