Answered: 18.6.2. Let R be a local commutative… | 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

Section6.4: Maximal Ideals (optional)

Problem 27E: 27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime...

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18.6.2. Let R be a local commutative ring with identity, and assume that the
maximal ideal in R is a principal ideal. Let a be an irreducible element
of R. Show that (a) is maximal.

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Exercises If and are two ideals of the ring , prove that is an ideal of .
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.
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 .
Show that the ideal is a maximal ideal of .
17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic 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 .)
27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.
Prove that if R is a field, then R has no nontrivial ideals.
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 ].
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
. 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 .

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