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 divisor.

Q: Show that a unit of a ring divides every element of the ring.

A: Let us assume a unit in a ring R as a. Let us assume any element as x in that ring R.  By using the…

Q: Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that…

A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

Q: (a) Let R be a ring and S a subset of R. What does it mean to say that S is a subring of R?

A: a. S is a subset of R. A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S. A subring…

Q: if in a ring R every x in R satisfies x^2=x , prove that R must be commutative

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Q: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

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Q: а. Let a, b, c be three elements of a Euclidean ring R and (a, b) = 1. If a/bc then show that a/c.

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Q: Let R be a ring with unity 1 and char (R) = 3. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=3.  Then R contains a subring isomorphic to_______.

Q: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: Give an example of a non-commutative ring R without unity such that (xy)^2 = x^2.y^2 for all x,y in…

A: We consider the example of a non-commutative ring R without unity such that (xy)2 = x2y2 for all x…

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: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: Show that a ring is commutative if it has the property that ab = caimplies b = c when a ≠ 0.

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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: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring True O…

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

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Q: Let 1={0,2} CZ. Prove if ring Z,/l is isomorphic with a ring Z,

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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) = { E xay : x, y e R }. finite

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Q: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

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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: Find elements a, b, and c in the ring Z ⨁ Z ⨁ Z such that ab, ac, andbc are zero-divisors but abc is…

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

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Q: Let R and S be rings and let ø : R → S be a ring homomorphism. Show that o is one-to-one if and only…

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

Q: Let R be a commutative ring and a e R. Show that if a is nilpotent, then ab is nilpotent for each be…

A: Solution. Since a is nilpotent , an = 0 for some n ∈ N.

Q: If R is commutative ring then deg(f(x) + g(x)) = max(deg(f(x)),deg(g(x)). %3D

A: We have to solve given problem:

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: Let R be Euclidean ring and a, b non - Zevo elements of R- If dca) <d lab), are show that b is not…

A: Let a and b be non-zero elements of R. Assume that b is a unit in R. Then, for some element c in R…

Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

Q: If R is a commutative ring with no zero divisors then deg(f(x).g(x)) = deg ƒ(x)+ deg g(x). True…

A: We use definition of non zero divisors.

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: 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: be a Euclidean ring and asb elements of R- gf a and b Let R are non- ze YO asso ciates , Show that…

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Q: Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca,  then b=c.

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Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

Q: Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 =…

A: Given Suppose that R is a ring with nonzero-divisors and that R contains a nonzero element b such…

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: If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.

A: R1 and R2 are subrings of the ring R, prove that R1∩R2 is a subring of R

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

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Q: Determine whether (Z,⊕,∗) is a ring with binary operations defined as (x⊕y)=x+y−7,…

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Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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Q: Let A, B,C be rings. Let o be a ring homomorphism from A into B and B be a ring homomorphism from B…

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Q: Let R be a ring. Let p= ax + b and q = cx² + dx + e be elements of R[x], for a, b, c, d, e E R. True…

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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

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Q: Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides…

A: Let a, b, and c be elements of a commutative ring where a is a unit. Suppose that bdivides c. Then c…

Q: The ring 3z is isomorphic to the ring 5z True False

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

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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.