Only true or false 4) Mark True or False each of the followinga) The idempotents Of (Z6,0,0 ,) are ONLY 0, 1, andb) The number 161 is an irreducible element in Z[i]c) A local Ring is a Ring with only One Prime ideald) The ideal < 0> is a prime ideal in (Z₁2,,,).e) If u E I where I is an Ideal in R and u is a unit Then I =f) If R is an Integral domain, then the right cancellation law hol

As per the company rule, we are supposed to solve the first three sub-parts of a multi-parts…

Answered: a) The idempotents Of (Z6,0,0,) are…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.4: Maximal Ideals (optional)

Problem 13E

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Exercises If and are two ideals of the ring , prove that is an ideal of .
. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
14. Let be an ideal in a ring with unity . Prove that if then .
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .
Prove that if R is a field, then R has no nontrivial ideals.
An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.
Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
8. Prove that the characteristic of a field is either 0 or a prime.
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 .)

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a) The idempotents Of (Z6,0,0,) are ONLY 0, b) The number 161 is an irreducible element in Z[i] c) A local Ring is a Ring with only One Prime ideal