Homework Answers
This problem is about annihilator of an ideal. In this problem annihilation of sum of two ideals is intersection of annihilatior of individual. Also it says that annhilator of intersection of two ideal need not equal to sum of annhilator of individual
Additional Problems: Let R be a commutative unital ring, and let M be an R-module. (1)...
Additional Problems: Let R be a commutative unital ring, and let M be an R-module. (1) The set (reRIrm 0 for all m e M) is called the annihilator of M. Show that it is an ideal of R. (2) Show that the intersection of any nonempty collection of submodules of M is a sub- module.
Prove/Justify. help plz. Remark 8.46. The following facts are easily verified. (a) (A) is the intersection of all ideals containing A. (b) If R is commutative, then (a)-aR :-|ar l r є R. Example 8.47...
Prove/Justify. help plz.
Remark 8.46. The following facts are easily verified. (a) (A) is the intersection of all ideals containing A. (b) If R is commutative, then (a)-aR :-|ar l r є R. Example 8.47. In Z, nZ = (n) = (-n). In fact, these are the only ideals in Z (since these are the only subgroups). So, all the ideals in Z are principal. If m and n are positive integers, then nZ C mZ if and only if...
Let R be a commutative ring which has exactly four ideals {0}, I, J, and R....
Let R be a commutative ring which has exactly four ideals {0}, I, J, and R. Among all such rings find a ring which has the smallest number of elements.
(7) Let R be a ring with 1 and let M be a unital left R-module....
(7) Let R be a ring with 1 and let M be a unital left R-module. If I is a right ideal of R then the annihilator of I in M is defined to be AnnM(I) = {m € M: am=0 for all a € 1}. (a) Prove that Annm(I) is a submodule of M. (b) Take R = Z and M = Z/3Z Z/102 x Z/4Z. If I = 2Z describe AnnM(I) as a direct product of cyclic groups.
Abstract Algebra (1) Let I, J C R be ideals. Show that if I is generated...
Abstract Algebra
(1) Let I, J C R be ideals. Show that if I is generated by n elements, and J is generated by m elements, then I +J is generated by no more than nm elements. 1
Let R be Commutative ring with 1 and let N and M be two R-modules Prove...
Let R be Commutative ring with 1 and let N and M be two R-modules Prove that NM MBN
Let R be Commutative ring with 1 and let N and M be two R-modules Prove that NM MBN
3. [10] Let L and M be left ideals of a ring R. Prove that L...
3. [10] Let L and M be left ideals of a ring R. Prove that L + M 2EL,Y EM} is a left ideal of R. = (x+y
13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be...
13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,...,...
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that...
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,...,...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
Additional Problems: Let R be a commutative unital ring, and let M be an R-module. (1) The set (reRIrm 0 for all m e M) is called the annihilator of M. Show that it is an ideal of R. (2) Show that the intersection of any nonempty collection of submodules of M is a sub- module.
Prove/Justify. help plz.
Remark 8.46. The following facts are easily verified. (a) (A) is the intersection of all ideals containing A. (b) If R is commutative, then (a)-aR :-|ar l r є R. Example 8.47. In Z, nZ = (n) = (-n). In fact, these are the only ideals in Z (since these are the only subgroups). So, all the ideals in Z are principal. If m and n are positive integers, then nZ C mZ if and only if...
(7) Let R be a ring with 1 and let M be a unital left R-module. If I is a right ideal of R then the annihilator of I in M is defined to be AnnM(I) = {m € M: am=0 for all a € 1}. (a) Prove that Annm(I) is a submodule of M. (b) Take R = Z and M = Z/3Z Z/102 x Z/4Z. If I = 2Z describe AnnM(I) as a direct product of cyclic groups.
Abstract Algebra
(1) Let I, J C R be ideals. Show that if I is generated by n elements, and J is generated by m elements, then I +J is generated by no more than nm elements. 1
Let R be Commutative ring with 1 and let N and M be two R-modules Prove that NM MBN
Let R be Commutative ring with 1 and let N and M be two R-modules Prove that NM MBN
3. [10] Let L and M be left ideals of a ring R. Prove that L + M 2EL,Y EM} is a left ideal of R. = (x+y
13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,...,...
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
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