Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the overlying and M⊆R is maximal ideal, then ϕ (M)⊆S is maximal ideal.

Q: The function p: Zs → Z30 defined by ø(a) = 6a is a ring homomorphism. True False

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Q: Let R be a commutative ring with unity, and let I be a proper idealwith the property that every…

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Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then... ..... .... (a) 1 = R (b)…

A: {R, +, .) is a ring and it is given that I is not a maximal ideal in R.From this, we conclude that I…

Q: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

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Q: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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

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Q: If fis a ring homomorphism from Zm to Z, such that f(1)= b, then b**2 = b*. True False

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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

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Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

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Q: Let Roll no be a ring with ideals I and J , such that I ⊆ J . Then J/I is an ideal of Roll no/I .

A: Ideal: A non-empty subset I of  a ring R is called ideal in a ring R if following conditions holds:…

Q: If Ø: R → S is a ring homomorphism. The Ø preserves: All of these Nilpotent elements Units O…

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Q: For a fixed element a of a ring R, prove that the set {x ϵ R I  ax = O} is a subring of R.

A: Given :            The ring R and a fixed element 'a' of R.   To prove that the set x ∈ R | ax =…

Q: (1) For every ring R and R-module M below, determine whether M 0 and prove your answer. (a) R= Z, M…

A: We evaluate elementary tensors and prove that they are 0.

Q: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

Q: element a ∈ R, define the Annihilator of a denoted as Ann(a), as Ann(a) = {r ∈ R| r.a = 0} Show that…

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Q: The ring Zs[i] has no proper ideals

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Let R be a commutative ring with identity. Using the homomorphism theorem (Theorem 16.45) and…

A: Recall that in a ring A  not necessarily commutative and with an identity, an ideal M⊂A is a maximal…

Q: Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an…

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Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J…

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Q: Suppose R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

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Q: Let R be a commutative unitary ring and let M be an R-module. For every r ERlet rM = {rx; x E M} and…

A: We shall solve first question only as you have asked more than one different question as per company…

Q: Let R be a commutative ring with unity and let a ∈ R be fixed. Prove that the subset Ia = {x ∈ R :…

A: Given below the detailed solution

Q: If fis a ring homomorphism from Zm to Z, such that f (1) = b, then b*+2 = b*. True False

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Q: Is the mapping from Z5 to Z30 given by x --> 6x a ring homomorphism?Note that the image of the…

A: Given a mapping,

Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

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Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

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Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the covering and M⊆R is maximal ideal, then…

A: Given φ:R→S be a ring homomorphism. Let,, φ is covering and M⊂R is maximal ideal.  To prove that…

Q: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…

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Q: a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that…

A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…

Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: '. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A: R is ring with unity and I is ideal of R.

Q: Given a commutative ring R considered as a module over itself, determine all the module…

A: Given: A commutative ring R is considered as a module over itself.   To determine: All the module…

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

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Q: Let f: MM' be an R-module homomorphism, then

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Q: Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

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Q: The ring Zs[i] has no proper ideals True False O O

A: We check whether Z8[I] has proper ideal.

Q: Iff is a ring homomorphism from Zm to Zn such that f (1)=b, then b*+2 = b*. False True

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Q: If U, V are ideals of a ring R, let U + V = {u+ v:u E U,v E V}. Prove that U +V is also an ideal.

A: We have to prove the conditions of ideal

Q: 2) Let M be a noetherian R-module over commutative ring R. If f : M M is onto ho- momorphism , show…

A: Please check step 2 and 3 for solution.

Q: Given that (I,+,.) is an ideal of the ring (R,+,.) Show that : a- the ring (R/I,+,.) may have…

A: We have to show that by example

Q: Theorem 12. Let R is a commutative ring and r e R,fe Hom (M, M'), then rfe. R M, M') defined by (rf)…

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Q: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

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Q: If Ø: R → S is a ring homomorphism. The Ø preserves: Units All of these Nilpotent elements…

A: The units in a ring are those elements which have an inverse under multiplication. An element x of…