Let R=(m+n√2|m,n∈z) and let I=(m+n√2|m,n∈Z,and m is even) Find the well known commutative ring to which R/I is isomorphic.?
Let R=(m+n√2|m,n∈z)and let I=(m+n√2|m,n∈Z,and m is even)
(a) show that I is an ideal of R. (b) Find the well known commutative ring to which R/I is isomorphic. Hint: How many congruence classes does I determine?
To show that I is an ideal of R I need to show that I is closed under addition and subtraction under multiplication by a member of R. Which is fairly simple. For part b I believe that R/I is isomorphic to Z2 but I have no idea how construct the isomorphism.
Any help is appreciated!
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You are correct about part a.
As for part b, consider the ring homomorphism f: R → Z₂ defined by
f(m + n√2) = m (mod 2).
It is easy to check that f is a ring homomorphism.
For any m + n√2, r + s√2 in R (with m,n,r,s being integers), we have
f((m + n√2) + (r + s√2))
= f((m + r) + (n + s)√2)
= m + r (mod 2)
= f(m + n√2) + f(r + s√2)
f((m + n√2)(r + s√2))
= f(mr + 2ns + (ms + nr)√2)
= mr + 2ns (mod 2)
= mr (mod 2)
= f(m + n√2) f(r + s√2).
f is onto, because f(0) = 0 (mod 2) and f(1) = 1 (mod 2)
(and 0, 1 are in R).
Finally, we check the kernel.
ker f = {a + b√2 in R : f(a + b√2) = 0 (mod 2)}
........= {a + b√2 in R : a = 0 (mod 2)}
........= {a + b√2: a is even (and a, b are integers)}
........= I.
Hence, R/I ≅ Z₂ by the First Isomorphism Theorem.
I hope this helps!