Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse of pqr.

Q: Let R[x] be the set of all polynomials over a ring (R ,+, . ). Then, (R[x] ,+ , .) form ring , which…

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Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

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Q: Let R be a ring and let I be an ideal of R. Prove that the factor ring R/I is commutative iff rs-sr…

A: We have to prove that factor ring R/I is commutative iff rs-sr is in R for all r and s in R.

Q: Let R be a ring with unity. Show that (a) = { £ xay : x, y e R }. finite

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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: Consider the ring R= [r.s.t) whose addition and multiplications ubles are given below, Then t.s= S

A: A ring in the mathematical sense is a set S together with two binary operators + and * which…

Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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Q: Let R be a ring with identity 1 and let a be an element of R such that a2 = 1. Let S = { ara : r e…

A: we will use sub ring test.

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 R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

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Q: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

Q: Let R be a ring with 1 0. Prove or disprove: (a) if R has no ideals other than {0} and R, then R is…

A: Given statement is false. Justification is in step 2

Q: Let R be a ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…

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Q: a is a unit in a ring R with unity, then a is not a zero divisor

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Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

Q: The ring Z is isomorphic to the ring 3Z O True False

A: Solution:

Q: Let R be a ring with unity. Show that (a) = { £ xay: x, y e R }. finite

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Q: Let R be a ring with unit. Define 2R ≔ 1R + 1R. In most rings, 2R ≠ 1R. Figure out what the…

A: Please check the answer in next step

Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

A: Ring (definition) Let R be a non empty set together with two binary operations called addition(+)…

Q: Let R be a ring with unity, n a positive integer and a, b e R. Prove: If ab = ba, then (a + b)"…

A: Mathematical Induction  Let us consider a statement P(n) 1. Prove the given statement form n=1…

Q: = Let I andJ be two ideals of a commutative ring R. Show that: S= {r€R: ri E J Vi EI } is an ideal…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

Q: Let R be a commutative ring with unity. Fix two elements a, b ∈ R. Prove that if a = bt for some t ∈…

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Q: Given that I = {(a,b) E R | a and b are even numbers} is an ideal of a ring R = Z × Z, construct…

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Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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Q: Let R be a commutative ring with identity and let I be a proper ideal of R. Prove that R/I is a…

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Q: Let R be ring, then R is T F imbedded in the polynomial ring R[X].

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Q: Is the ring Z/6Z and Z6 are isomorphic.

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

A: The complete solutions are given below

Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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Q: Let R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.

A: Solution

Q: Prove that R[x2, x3] is a subring of the ring R[x].

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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: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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Q: If R is a commutative ring, shw that the characteristic of R[x] is the same as the characteristic of…

A: Given: R is commutative ring

Q: Consider the ring (Q,Ð,0), where the "addition" O and "multiplication" ® are defined as follows:…

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Q: Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is…

A: Given R be a commutative ring with unit element. If f(x) is a prime ideal of R[x] then we have to…

Q: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let R be a ring and M be an R-module. Ther

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Q: Let A be an ideal of a ring R. i) If R is commutative then show that R/A is commutative ii) If R…

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Q: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.

A: We are given that R be a commutative ring with unity. We have to show that  R is a field if and only…

Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…