Answered: 18.6.1. Let R be a non-trivial… | bartleby

Elements Of Modern Algebra

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...

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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.

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Exercises If and are two ideals of the ring , prove that is an ideal of .
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
19. Find a specific example of two elements and in a ring such that and .
Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.
Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.

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