Let I ⊆ R be an ideal. (a) Prove that every element of R/I is a solution of x2 = x if and only if r2 − r ∈ I for all r ∈ R. Is R/I an integral domain? (b) Suppose that R is an integral domain. Is R/I necessarily an integral domain? If so, prove it. If not, provide a counterexample.

Let I ⊆ R be an ideal. (a) Prove that every element of R/I is a solution of x2 = x if and only if r2 − r ∈ I for all r ∈ R. Is R/I an integral domain? (b) Suppose that R is an integral domain. Is R/I necessarily an integral domain? If so, prove it. If not, provide a counterexample.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 16E: Prove that if a subring R of an integral domain D contains the unity element of D, then R is an...

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Question

Let I ⊆ R be an ideal.
(a) Prove that every element of R/I is a solution of x² = x if and only if r² − r ∈ I for all r ∈ R.
Is R/I an integral domain?
(b) Suppose that R is an integral domain. Is R/I necessarily an integral domain? If so, prove it.
If not, provide a counterexample.

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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