Let (R,m) be a commutative local ring. Recall that this means that R is a commutative ring with unique maximal ideal m. Prove that the following conditions are equivalent: a) If M is a finitely generated R-module and mM = M, then M = 0. b) If M is a finitely generated R-module, N containing M is a submodule, and N + mM = M, then N=M. c) If M is a finitely generated R-module and {m1,....,mn} in M are elements of M whose images {n1,....nn} form a generating set for the R-module M/mM, then {m1,....,mn} form a generating set for M.

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Q: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.

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Q: Let R be a ring with unity which has exactly one maximal left ideal M. Show that the only…

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Q: Let R be a ring and fix b ∈R. (a) Show that S = {n1R ∈R : n ∈Z } is a subring of R. (b) Show that T…

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Q: Let R be a commutative ring with identity such that every submodule of every free module is free.…

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Q: Let R be a commutative ring with unity and let S ‡ R be an ideal of R. Then R /S is an integral…

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Q: Q) Put True (T) or Fuls (F) for the following: (1)Every prime ideal I of ring R is maximal ideal.…

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Q: Let R be a ring and let I a minimal left ideal  of R. If M is a faithful simple left R-mod

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Q: Prove that in a Euclidean ring R, (a, b) can be found as follows : b = q0 a + r¡, where d (r¡)<d (a)…

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Q: 1. Let M be a proper ideal in a Boolean ring R with unity. Prove that (i) R/M is a Boolean ring and…

A: It is given that, M is a proper ideal in a Boolean ring R with unity.

Q: Let R be a commutative ring with an identity 1R and let J be a proper ideal with the property that…

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Q: 1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes…

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Q: Given the system of equations: ax + by + cz = 1 ex + dy + fz = -2 9x+2y+-17z = 0 and considering…

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Q: unity. Show, by example, that a maximal ideal in R need not be a maximal id

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Q: If A is a commutative ring with unity, show that every prime ideal is also maximal.

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Q: Let & be a ring with the invariant rank property and let M and N be free R-modules. Then M IN ift…

A: R be a ring with the invariant rank propertyand let M and N be free R-modules.

Q: Let R be a Boolean Ring. If {0} and R are the only ideals of R, prove that R is isomorphic to Z₂-

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