(1) Let a be a fixed element in the commutative ring R, and let A- (xe R: xa = 0). Prove that A is an ideal of R. (2) Suppose that R and R, are rings and DR R₁ is a ring epimorphism and that K₁, K, are two distinct ideas of R₁. Let Jxe R: 0(x) = K₁).J₂ (x = R: 0(x) = K₂), be the ideals of R. Show that J₁ J₁. E (3) Let R denote a commutative ring with a one. An element x of Ris termed nilpotent if and only 3 e N such x" - 0. (i) Give an example of a nilpotent element in Z- (ii) Show that if J is an ideal of R and x is a nilpotent element of R, then x+Jis a nilpotent element of R/J. (ii) Show that if X is an ideal of R and if all the elements of K are nilpotent and all the elements of the ring R/K are nilpotent, then all the elements of R are nilpotent. (iv) Suppose R is commutative. Show that the set N of all nilpotent elements in R is an ideal of R and that R/N contains no nonzero nilpotent elements. (v) Suppose R is commutaive. Using (ii), show that N is contained in every maximal ideal of R. (4) Let / be an ideal of the ring R. Show that the ring R// is commutative if and only if xy-yx e J for every x.y R. Deduce that if X₁ and X₂ are ideals of R and both R/K, and R/K, are commutative, then R/(KK) is also commutative. (5) Suppose that D is an integral domain and that J and K are ideals of D neither of which equals (0). Show that JK (0). (6) Let R be a commutative ring with a one, and let / be an ideal of R. Show that if R/J is a field, then / is a maximal ideal. (7) Let R be a finite commutative ring with unity. Show that every prime ideal of R is a maximal ideal.. (8) Show that for a field F, the set S of all all matrices of the form a b (0) 00 for a, b F is a right ideal but not a left ideal of M₂ (F). (9) Let A and B be ideals of a commutative ring R. The quotient A B of A4 by B is defined by A: B=(reR: rb e A for all be B. Show that A B is an ideal of R. (10) Show that : CM₂(R) given by for a, b R is a homomorphism. (11) Let p = Z be prime. Show that (i) pZ is a maximal ideal. a b - (+8) -ba (a + ib) =
(1) Let a be a fixed element in the commutative ring R, and let A- (xe R: xa = 0). Prove that A is an ideal of R. (2) Suppose that R and R, are rings and DR R₁ is a ring epimorphism and that K₁, K, are two distinct ideas of R₁. Let Jxe R: 0(x) = K₁).J₂ (x = R: 0(x) = K₂), be the ideals of R. Show that J₁ J₁. E (3) Let R denote a commutative ring with a one. An element x of Ris termed nilpotent if and only 3 e N such x" - 0. (i) Give an example of a nilpotent element in Z- (ii) Show that if J is an ideal of R and x is a nilpotent element of R, then x+Jis a nilpotent element of R/J. (ii) Show that if X is an ideal of R and if all the elements of K are nilpotent and all the elements of the ring R/K are nilpotent, then all the elements of R are nilpotent. (iv) Suppose R is commutative. Show that the set N of all nilpotent elements in R is an ideal of R and that R/N contains no nonzero nilpotent elements. (v) Suppose R is commutaive. Using (ii), show that N is contained in every maximal ideal of R. (4) Let / be an ideal of the ring R. Show that the ring R// is commutative if and only if xy-yx e J for every x.y R. Deduce that if X₁ and X₂ are ideals of R and both R/K, and R/K, are commutative, then R/(KK) is also commutative. (5) Suppose that D is an integral domain and that J and K are ideals of D neither of which equals (0). Show that JK (0). (6) Let R be a commutative ring with a one, and let / be an ideal of R. Show that if R/J is a field, then / is a maximal ideal. (7) Let R be a finite commutative ring with unity. Show that every prime ideal of R is a maximal ideal.. (8) Show that for a field F, the set S of all all matrices of the form a b (0) 00 for a, b F is a right ideal but not a left ideal of M₂ (F). (9) Let A and B be ideals of a commutative ring R. The quotient A B of A4 by B is defined by A: B=(reR: rb e A for all be B. Show that A B is an ideal of R. (10) Show that : CM₂(R) given by for a, b R is a homomorphism. (11) Let p = Z be prime. Show that (i) pZ is a maximal ideal. a b - (+8) -ba (a + ib) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do no.10
![(1) Let a be a fixed element in the commutative ring R, and let / -
(xe R: xa= 0). Prove that A is an ideal of R.
(2) Suppose that R and R, are rings and DR
two distinct ideas of R₁. Let Ji
ideals of R. Show that J₁ J₁.
R₁ is a ring epimorphism and that K₁, K, are
(x = R: 0(x) = K₁).J₂ (x R: 0(x) = K₂), be the
E
(3) Let R denote a commutative ring with a one. An element x of Ris
termed nilpotent if and only 3 e N such x" - 0.
(i) Give an example of a nilpotent element in Z-
(ii) Show that if J is an ideal of R and x is a nilpotent element of R, then
x+Jis a nilpotent element of R/J.
(iii) Show that if K is an ideal of R and if all the elements of K are nilpotent and all the
elements of the ring R/K are nilpotent, then all the elements of R are nilpotent.
(iv) Suppose R is commutative. Show that the set N of all nilpotent elements in R is an
ideal of R and that R/N contains no nonzero nilpotent elements.
(v) Suppose R is commutaive. Using (ii), show that N is contained in every maximal ideal
of R.
(4) Let J be an ideal of the ring R. Show that the ring R/J is commutative
if and only if xy-yx € / for every x.y R. Deduce that if X₁ and
X₂ are ideals of R and both R/K, and R/K, are commutative,
then R/(KK) is also commutative.
(5) Suppose that D is an integral domain and that J and K are ideals
of D neither of which equals (0). Show that JK (0).
(6) Let R be a commutative ring with a one, and let / be an ideal of R. Show that if R/J is a
field, then / is a maximal ideal.
(7) Let R be a finite commutative ring with unity. Show that every prime ideal of R is a
maximal ideal.
(8) Show that for a field F, the set S of all all matrices of the form
a b
(8)
for a, b F is a right ideal but not a left ideal of M₂ (F).
(9) Let A and B be ideals of a commutative ring R. The quotient A B of A4 by B is defined by
A: B= (reR: rb e A for all be B.
Show that A B is an ideal of R.
(10) Show that : CM₂(R) given by
ab
- (88)
-ba
(a+ib) =
for a, b R is a homomorphism.
(11) Let p = Z be prime. Show that
(i) pZ is a maximal ideal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c044b69-454e-4f9e-8cd6-567ae6231ec2%2F80cbf44a-84b2-4e5c-a0bb-6f10ecb987f8%2Fwilbslf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(1) Let a be a fixed element in the commutative ring R, and let / -
(xe R: xa= 0). Prove that A is an ideal of R.
(2) Suppose that R and R, are rings and DR
two distinct ideas of R₁. Let Ji
ideals of R. Show that J₁ J₁.
R₁ is a ring epimorphism and that K₁, K, are
(x = R: 0(x) = K₁).J₂ (x R: 0(x) = K₂), be the
E
(3) Let R denote a commutative ring with a one. An element x of Ris
termed nilpotent if and only 3 e N such x" - 0.
(i) Give an example of a nilpotent element in Z-
(ii) Show that if J is an ideal of R and x is a nilpotent element of R, then
x+Jis a nilpotent element of R/J.
(iii) Show that if K is an ideal of R and if all the elements of K are nilpotent and all the
elements of the ring R/K are nilpotent, then all the elements of R are nilpotent.
(iv) Suppose R is commutative. Show that the set N of all nilpotent elements in R is an
ideal of R and that R/N contains no nonzero nilpotent elements.
(v) Suppose R is commutaive. Using (ii), show that N is contained in every maximal ideal
of R.
(4) Let J be an ideal of the ring R. Show that the ring R/J is commutative
if and only if xy-yx € / for every x.y R. Deduce that if X₁ and
X₂ are ideals of R and both R/K, and R/K, are commutative,
then R/(KK) is also commutative.
(5) Suppose that D is an integral domain and that J and K are ideals
of D neither of which equals (0). Show that JK (0).
(6) Let R be a commutative ring with a one, and let / be an ideal of R. Show that if R/J is a
field, then / is a maximal ideal.
(7) Let R be a finite commutative ring with unity. Show that every prime ideal of R is a
maximal ideal.
(8) Show that for a field F, the set S of all all matrices of the form
a b
(8)
for a, b F is a right ideal but not a left ideal of M₂ (F).
(9) Let A and B be ideals of a commutative ring R. The quotient A B of A4 by B is defined by
A: B= (reR: rb e A for all be B.
Show that A B is an ideal of R.
(10) Show that : CM₂(R) given by
ab
- (88)
-ba
(a+ib) =
for a, b R is a homomorphism.
(11) Let p = Z be prime. Show that
(i) pZ is a maximal ideal.
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