18.6.1. Let R be a non-trivial commutative ring with identity. Show that the following are equivalent: (a) R is local. (b) The set of all non-units of R forms an ideal of R. (c) The sum of any two non-units in R is a non-unit. (d) If x ЄR, then x or 1 - x is a unit.
18.6.1. Let R be a non-trivial commutative ring with identity. Show that the following are equivalent: (a) R is local. (b) The set of all non-units of R forms an ideal of R. (c) The sum of any two non-units in R is a non-unit. (d) If x ЄR, then x or 1 - x is a unit.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 24E: 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set...
Related questions
Question
![18.6.1. Let R be a non-trivial commutative ring with identity. Show that the
following are equivalent:
(a) R is local.
(b) The set of all non-units of R forms an ideal of R.
(c) The sum of any two non-units in R is a non-unit.
(d) If x ЄR, then x or 1 - x is a unit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5fbaae5-8d47-4476-8095-8b380294ae7e%2F4529d67b-c801-495f-aa1a-7dd54a9cc0bf%2Fqv6k3py_processed.png&w=3840&q=75)
Transcribed Image Text:18.6.1. Let R be a non-trivial commutative ring with identity. Show that the
following are equivalent:
(a) R is local.
(b) The set of all non-units of R forms an ideal of R.
(c) The sum of any two non-units in R is a non-unit.
(d) If x ЄR, then x or 1 - x is a unit.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,