An element x of a ring R is idempotent if x2 = x. (a) Find four idempotent elements of GL(2, R (this R represents real numbers) ). (b) Find all idempotents in Z12. (c) Prove that if every element of R is idempotent, then R is commutative.

An element x of a ring R is idempotent if x2 = x. (a) Find four idempotent elements of GL(2, R (this R represents real numbers) ). (b) Find all idempotents in Z12. (c) Prove that if every element of R is idempotent, then R is commutative.

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

An element x of a ring R is idempotent if x² = x.
(a) Find four idempotent elements of GL(2, R (this R represents real numbers) ).
(b) Find all idempotents in Z₁₂.
(c) Prove that if every element of R is idempotent, then R is commutative.

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