Answer to Question #145988 in Abstract Algebra for Sourav Mondal

Question #145988
Let R = Z + √2 Z and S = the ring of 2 x 2
matrices of the form [
a 2b
b. a
. ] Where a,b belongs to Z

Show that f : R --> S defined by
f (a + √2 b) = [
a. 2b
b. a
] is an isomorphism of rings.
1
Expert's answer
2020-11-24T16:53:11-0500

Surjectivity is obvious since S is the set of matrices of that special form.

Injectivity: . Hence, equating the columns of the matrices we get, Hence injective.

Homomorphism: Also Also . Hence homomorphism


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