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(3) Let R denote a commutative ring with a one. An element x of Ris termed nilpotent if and only 3n 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 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 J be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx e J for every x,y R. Deduce that if K, and Ky are ideals of R and both R/K, and R/K, are commutative, then R/(K, K₂) 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 J be an ideal of R. Show that if R/J is a field that

(3) Let R denote a commutative ring with a one. An element x of Ris termed nilpotent if and only 3n 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 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 J be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx e J for every x,y R. Deduce that if K, and Ky are ideals of R and both R/K, and R/K, are commutative, then R/(K, K₂) 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 J be an ideal of R. Show that if R/J is a field that

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(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.
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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