Let R and S be rings.a. Show that the mapping from R ⨁ S onto R given by (a, b) --> a is a ring homomorphism.b. Show that the mapping from R to R ⨁ S given by a --> (a, 0) is a one-to-one ring homomorphism.c. Show that R ⨁ S is ring-isomorphic to S ⨁ R.

Let R and S be rings.a. Show that the mapping from R ⨁ S onto R given by (a, b) --> a is a ring homomorphism.b. Show that the mapping from R to R ⨁ S given by a --> (a, 0) is a one-to-one ring homomorphism.c. Show that R ⨁ S is ring-isomorphic to S ⨁ R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let R and S be rings.
a. Show that the mapping from R ⨁ S onto R given by (a, b) --> a is a ring homomorphism.
b. Show that the mapping from R to R ⨁ S given by a --> (a, 0) is a one-to-one ring homomorphism.
c. Show that R ⨁ S is ring-isomorphic to S ⨁ R.

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