12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only if every zero divisor of the quotient ring (R/I,+,') is nilpotent.

Q: identity an let N denote the Bet ol nilpotent elements of R. Verify that a) the triple (N,+,) is an…

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Q: Exercises :- 1- Show that: (Z24 / {0,6, 12 ,18},+,.) = (Z6, +6 r6) 2- Show that: (Z20 /{0,10},+,.)…

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Q: Define principal ideal ring.

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A: Given that x^4=x for any x in R. To prove R is commutative.

Q: 2- Let f be an isomorphism from the ring (R, +,) to the ring (R', +','). If (I, +;) is an ideal of…

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Q: How many distinct cosets are there in the factor ring Z[i] / ? Explain your answer.

A: Given: Let's determine the number of distinct cosets of the factor ring ℤi3+i.

Q: 7. If f is a homomorphism from the ring (R, -+, ) onto the ring (R', +', '), show the following: a)…

A: Given that f is a homomorphism from the ring (R,+,.) onto the ring (R',+',.') The objective is to…

Q: Let (R,+,.) be acommutive ring with identity and a∈R be an idempotent which is different from 0 or 1…

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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: 2- An example of two ideals A and B of the ring (Z12. +2-u) such thata B are: (a) A and B (c) A- and…

A: First we find all subgroup of Z12

Q: ísomorp

A: Isomorphic to quotient ring:

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Q: The ring Z,3 has exactly-------------maximal ideals

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Q: Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition…

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

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Q: Q3: Prove that the ring of rational numbers (Q, +,.) is division ring

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Q: Find the characteristic of the ring (Z/20Z) /(5Z/20Z)

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Q: Prove. Let f : R → R' be a ring homomorphism. If I' is an ideal of R', then f^-1(I') is an ideal of…

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Q: Q) Let (H,+10, -10) be a subring of the ring (Z10, +10. -10 ), where H = {[0], [2], [4], [6], [8]}.…

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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…

Q: 17. Given that (I, 1.) in an ideal of the ring (R,+,), show that a) whenever (R,1,) is commutative…

A: Definition Two-sided Ideal of a ring : An ideal I of a ring R is called a two-sided ideal if it…

Q: Let R be the ring Z[√−5]. (a) Prove that I = (2, 1 + √−5), the ideal generated by those two…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…