Let S be a ring. Determine whether S is commutative if it has the following property: whenever ry = zx for some elements 0 + x, y, z E S then y = z. (Prove or give a counterexample.)

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 commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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

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

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…

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Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.

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Q: a 2b Let Z[√√√2] = {a+b√2 \a, beZ} and let H = { [ b Show that Z[√2] and Hare isomorphic as rings. a…

A: We will be solving Q2 as mentioned. Given that ℤ2=a+b2:a,b∈ℤ and H=a2bba: a,b∈ℤ. Let, R and R' be…

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: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

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Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I=I ∩ rad R

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Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

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: Define the binary operations e and O on Z by a e b = a +b – 1 and a O b = a + b – ab for all (a):…

A: We will use basic properties of integral domain and ring homomorphisms to solve both these parts.

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: 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: Let R be a commutative ring with identity and I be ideal of R. Then I is primary if and only if…

A: The statement is true.

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 unity. Show that (a) = { E xay : x, y e R }. finite

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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: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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

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Q: If A is a unitary ring define its centre to be CenA = {x €A; Vy EA) xy = yx}. %3D Show that CenA is…

A: Unitary ring is s ring with unity ie  A ring containing multiplicative identity.

Q: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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Q: Let R be a ring with a subring S: Prove or disprove: If a ∈ R is a unit, and a ∈ S, then a is also a…

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Q: Let R be a ring and a=a for all a'e R, Then commutative. prove that R is

A: First we notice that x3=x for all x∈ℝ, so that means 2x3=2x and thus 8x=8x3=2x and so 6x=0. Thus…

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: If u is finitely additive on a ring R; E, F eR show p(E) +µ(F) = u(EJ F)+u(En F)

A: Here, we need to write the union of E and F as union of disjoint subsets then use the properties of…

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 with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

Q: The set of all idempotents of the ring Z is اختر احدى الاجابات O (1,0) O (0,1) O (0,-1,1} O (1,-1)

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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: Let R be a ring with unity 1, and S = {n.1 : n E Z}.Then S'is Ra subring of Rnot a subring of

A: A non-empty set R with two binary operations addition(+) and multiplication(·) is called ring if it…

Q: If A and B are ideals in a ring R such that A n B = {0}, prove that for every a E A and b E B, ab =…

A: Explanation of the answer is as follows

Q: Let R be a ring and S a subring of R. Prove that if 1R ∈S, then S has unity and 1S =1R

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Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Let A be a ring with identity e and let a, b E A such that ab = e. Then ba is a Idempotent element…

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Q: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…

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: If u is finitely additive on a ring R; E, F eR show p(E) +u(F) = µ(B F)+µ(EnF)

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Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: a) Let R be a ring Ei a3 = a #aER %3D Prove that R is commutatve.

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

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: Prove that Q[x]/<x2 - 2> is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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Q: Let R be a commutative ring such that a^2 = a for all a ∈ R, then show that a+a = 0.

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Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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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: Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.

A: Proof. Let R be commutative. Then ab = ba for all a,b ∈ R.