Let
be the direct sum of the rings
and
, where
is the subring of
.
What is the zero element of R? What is the unity of R? Also write down all of the zero-divisors of R.
My Attempt:
I think by 'zero element of R', they mean the identity of the abelian group (R, +) which is (0,0).
I think the unity is the identity under multipication and I think it is (1,1).
Am I right?
For the last question: we know that nonzero elements r, s of R are called zero-divisors if rs = 0. So the elements of R are:
(0,0), (0,3), (0,6), (0,9), (0,12)
(1,0), (1,3), (1,6), (1,9), (1,12)
(2,0), (2,3), (2,6), (2,9), (2,12)
(3,0), (3,3), (3,6), (3,9), (3,12)
Let r=(a,b) and s=(c,d). Then rs= (a,b)(c,d)=(0,0) iff b=d=0 so we have 3 possibilities: {(1,0),(2,0),(3,0)}, and the only one which works is (2,0) (because 2.2 mod 4 =0). Therefore (2,0) is the only zero-divisor. Is this correct?