Homework Answers
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.
3. Let I be a left ideal of R and let (: R) reRRCI (a) : R) is an ideal of R. If l is regular, th...
3. Let I be a left ideal of R and let (: R) reRRCI (a) : R) is an ideal of R. If l is regular, then (: R) is the largest ideal of R that is contained in 1 (b) If I is a regular maximal left ideal of Rand AR/I, then (A(R). Therefore J(R) na:R), where /runs over all the regular maximal left ideals of R. Theorem 1.4. Let B be a subset of a left module A...
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,...,...
Thanks 6. Let R be a ring and a € R. Prove that (i) {x E...
Thanks
6. Let R be a ring and a € R. Prove that (i) {x E R | ax = 0} is a right ideal of R (ii) {Y E R | ya=0} is a left ideal of R (iii) if L is a left ideal of R, then {z E R za = 0 Vae L} is a two-sided ideal of R NB: first show that each set in 6.(i), (ii), (iii) above is a subring T ool of...
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
A1. Let M be an R-module and let I, J be ideals in FR (a) Prove...
A1. Let M be an R-module and let I, J be ideals in FR (a) Prove that Ann(I +J) -Ann(I) n Ann(J). (b) Prove that Ann(InJ)2 Ann(I) + Ann(J). (c) Give an example where the inclusion in (b) is strict. (d) If R is commutative ald unital and I, J are cornaximal (that is, 1 +J-(1)), prove that Ann(InJ) Ann(I)+Ann(J).
Only question 13 ,thx! 13. If M is a finitely generated module over the P.I.D. R,...
Only question 13 ,thx!
13. If M is a finitely generated module over the P.I.D. R, describe the structure of M/Tor(M). 14. Let R be a P.I.D. and let M be a torsion R-module. Prove that M is irreducible (cf. Exercises 9 to 11 of Section 10.3) if and only if M = Rm for any nonzero element me M where the annihilator of mis a nonzero prime ideal (p).
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a...
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a left T-module. Let e: (MN)* L M R (NB, L) be given by e (moon, e) = m (nol). Let m.con, mone MORN, and lEl. Prove e (lm, BR.) + (m₂ Ore), d)= e(m, on, d) + (mon, e). This is the proof I'm working on. I need to show the map I've defined (and which is defined towards the middle of the proof)...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal in R. Let 0 : R + R/I denote the natural projection homomorphism, and write ř = º(r) = r +I. (a) Show that the function Ø : Mn(R) + Mn(R/I) M = (mij) Ø(M)= M is a surjective ring homomorphism with ker ý = Mn(I). (b) Use Homework 11, Problem 2, to argue that M2(2Z) is a maximal ideal in M2(Z). (c) Show...
o Is the Z-module Z e Q injective 2 Let I be a simple right ideal...
o Is the Z-module Z e Q injective 2 Let I be a simple right ideal of aring R. Prove that Ia direct summand of Riff I I
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.
3. Let I be a left ideal of R and let (: R) reRRCI (a) : R) is an ideal of R. If l is regular, then (: R) is the largest ideal of R that is contained in 1 (b) If I is a regular maximal left ideal of Rand AR/I, then (A(R). Therefore J(R) na:R), where /runs over all the regular maximal left ideals of R. Theorem 1.4. Let B be a subset of a left module A...
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,...,...
Thanks
6. Let R be a ring and a € R. Prove that (i) {x E R | ax = 0} is a right ideal of R (ii) {Y E R | ya=0} is a left ideal of R (iii) if L is a left ideal of R, then {z E R za = 0 Vae L} is a two-sided ideal of R NB: first show that each set in 6.(i), (ii), (iii) above is a subring T ool of...
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
A1. Let M be an R-module and let I, J be ideals in FR (a) Prove that Ann(I +J) -Ann(I) n Ann(J). (b) Prove that Ann(InJ)2 Ann(I) + Ann(J). (c) Give an example where the inclusion in (b) is strict. (d) If R is commutative ald unital and I, J are cornaximal (that is, 1 +J-(1)), prove that Ann(InJ) Ann(I)+Ann(J).
Only question 13 ,thx!
13. If M is a finitely generated module over the P.I.D. R, describe the structure of M/Tor(M). 14. Let R be a P.I.D. and let M be a torsion R-module. Prove that M is irreducible (cf. Exercises 9 to 11 of Section 10.3) if and only if M = Rm for any nonzero element me M where the annihilator of mis a nonzero prime ideal (p).
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a left T-module. Let e: (MN)* L M R (NB, L) be given by e (moon, e) = m (nol). Let m.con, mone MORN, and lEl. Prove e (lm, BR.) + (m₂ Ore), d)= e(m, on, d) + (mon, e). This is the proof I'm working on. I need to show the map I've defined (and which is defined towards the middle of the proof)...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal in R. Let 0 : R + R/I denote the natural projection homomorphism, and write ř = º(r) = r +I. (a) Show that the function Ø : Mn(R) + Mn(R/I) M = (mij) Ø(M)= M is a surjective ring homomorphism with ker ý = Mn(I). (b) Use Homework 11, Problem 2, to argue that M2(2Z) is a maximal ideal in M2(Z). (c) Show...
o Is the Z-module Z e Q injective 2 Let I be a simple right ideal of aring R. Prove that Ia direct summand of Riff I I
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