Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is 0. If 1 has order n under addition , then the characteristic of R is n.

Q: (b) Let R = {3n : n e Z} and let U = {6n : n e Z}. Assume that R, with the standard the standard…

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Q: 3. Let R be a ring. Suppose a subset S of R is a ring under the induced ring operations of R. (a) If…

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Q: Let us define two new operations on R called ~ and ∗ by the following formulas. For real numbers a…

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Q: invertible

A: Given that: Let R be a ring with one and a, b two elements in R so that ab = ba Let (a-b) ^ n = 0 ,…

Q: (4)Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to…

A: Given that, K=Z12 and ideals I and J are such that I=([4])=4Z12={0,4,8} J=([6])=6Z12={0,6} Now,…

Q: (2) (C) Show that C = 6 ||abER} a, commutative ring with unity even if M₂ (R) is not. Finally, show…

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Q: which ring is Z_24/(3) isomorphic to? a. Z_2 b. Z_3 c. Z_4 d. Z_8 e. none of thr above

A: The given ring R is ℤ_24 = {0,1,2,...,22,23}, and the ideal is (3) = {0,3,6,...,21}

Q: Let R be a commutative ring with unit which is not an integral domain. Show that quotient and…

A: Question asks to show that if R is a commutative ring with a unit that is not an integral domain,…

Q: Let R be a commutative ring with unity, let a & R and let (a) be the principal ideal of R generated…

A: Given R is a commutative ring with unity and a is in R.

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

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Q: Consider the ideal N= of the ring Z24. How many elements does the quotient ring Z24/N have?

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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: Is it true or false that if I is a principal ideal in a commutative ring R with multiplicative…

A: Is it true or false that if  is a principal ideal in a commutative ring  with multiplicative…

Q: 16. Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z*; n2 1} Show…

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Q: Let R be a commutative ring with unity of characteristic 4. Compute and simplify (a + b) for a, b e…

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Q: a) LetRbe a ring. We define the radical ofRto be the left idealNwhich is the intersection of all…

A: a.A left ideal of a ring R is a subgroup of the additive group of R that is also a left R-module,…

Q: 1. Consider the ring Z₁2 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, with addition and multiplication…

A: The objective of this question is to find all the cosets of the ideal I = (4) in Z12 and compute the…

Q: . Consider the ring Z with ideal . Give the ring tables of Z/ .

A: Consider the ring  with ideal HereLet R = We know thatAndWe have to write the addition and…

Q: 6. Find two ideals I₁ and 12 of the ring Z such that a. I, U2 is not an ideal of Z. b. I₁ U2 is an…

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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 c ER be a fixed element. (a) Prove that the set A =…

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

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Q: , then A'(V) is a free K – module of rank

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Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ........ O…

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Q: (10) Let I = (4) be a principal ideal of integer %3| ring Z, Then I is. . ideal * Primary Prime O…

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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: '. 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: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

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Q: Let R be a ring. An element a E R is called nilpotent if there exists an integer n> 0 such that an…

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Q: If R be a commutative ring with identity and a ∈ R is idempotent different from 0 or 1. Prove that R…

A: Given that R is a commutative ring  with unity. Also, a∈R  is an idempotent element ⇒a2=a ( a≠0,1)…

Q: Exercise 44. Show that 1 and -1 are units in any ring R. Show that if R is a ring with 0 # 1 then 0…

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