SOLUTION: 1) Let R and S be rings and define R x S = {(r,s):rES} to be the set of all pairs (r,s) with r ER and s ES. Define furthermore the addition and multiplication in R x S componentwi

Question 25274: 1) Let R and S be rings and define R x S = {(r,s):rES} to be the set of all pairs (r,s) with r ER and s ES. Define furthermore the addition and multiplication in R x S componentwise by (r1,s1)+(r2,s2)=(r1+r2,s1+s2) and (r1,s1).(r2,s2)=(r1.r2,s1.s2)
a) show that R x S is a ring again.
b) show that if R and S both have an identity element then so does R x S.
c) show that R x S has non-zero, non-regular elements if R and S both are different from the zero-ring.
d) Can R x S ever be a field? Justify your answer!
Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
Hi,
I don't mind helping you but I'm not going to answer your entire question for you in 'ready to hand in' form. Tell me how far you've got, and where you're stuck and I'll be happy to help out with any specific problems.
To start you off let's look at part a. We need to show that

is a ring. For something to be a ring it must be an abelian group under addition, and multiplication must be distributive over addition.
I don't think these are too difficult to show, see how you get on and do get back to me with the bit you are stuck on.
Hope that helps,
Kev
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