Below are addition and multiplication tables for a set R Is

Below are addition and multiplication tables for a set R. Is it a commutative ring? What are its zero-divisors and units? a b c d a b c d a b c d a a b a b a b b a d c c d a b C d a b c d d d c b a

Solution

for commutative ring it has to satisfy A+B= B+A and A.B=B.A

so given are commutative.

Zero devisors are: suppose A and B are non zero numbers then A+B=a(here identity element in +) and A.B= a          ( Identity element in .)

Units are: suppose A and B are non zero numbers then A+B=a(here identity element in +) and A.B= a          ( Identity element in .)

So no zero devisors and no Units here happens