Let R = Z x Z = {(x,y)|x,y in Z} Show that Z x 2Z is maximal in R. Where, Z x 2Z = {(x,y)| x in Z, y in 2Z}
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quotient out by it you get $\displaystyle \mathbb{Z}_2$ which is a field, thus that ideal is maximal.
I know that Z/2Z = Z_2 (is a field) but how do you know that ZxZ/Zx2Z is a field?
Just take a look at the cosets. $\displaystyle (a,b)+(Z, 2Z)=(a+Z,b+2Z)=(Z,b+2Z)$. There are only two cases, if b is even, it is the 0, if b is odd, it is the other one. Thus it is a group with 2 elements, it is the finite field of order 2.
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