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...
Related questions
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
(b) Suppose that R is an integral domain. Is R/I necessarily an integral domain? If so, prove it.
If not, provide a counterexample.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning