An ideal A of a commutative ring R with unity is said to be finitely generated if there exist elements a,, az, ..., a, of A such that A = (a,, a, ..., a,). An integral domain R is said to satisfy the as- cending chain condition if every strictly increasing chain of ideals ICI,c.must be finite in length. Show that an integral domain R satisfies the ascending chain condition if and only if every ideal of R is finitely generated.

Q: 7. (a) Let be an increasing sequence of ideals in a ring R. Prove that the union of all these 1, is…

A: Both the sub-parts are solved below.

Q: Show that f:Z2 -> Z2 defined by f(n) = n2-n is a ring homomarphism.

A: The element in Z2 are {0,1} Now, f(0) = 02 - 0 = 0           f(1) = 12 - 1 = 0 Hence, f maps Z2 to…

Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

Q: Consider the ring ℤ12. List the elements of the following principal ideals: (i) ⟨4⟩  (ii) ⟨9⟩…

A: Given ring is ℤ12.

Q: Let R be a commutative ring with unity, I an arbitrary ideal in R and P1,P2,...,Pn prime ideals in…

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Q: ring R, an element a is a regular element if it is not a zero divisor in R. How many regular…

A: In general, an element a in a ring R is a zero divisor if there exists a nonzero element b in R such…

Q: If Z₁2 is a ring under addition and multiplication modulo 12then find its all prime ideals and…

A: If Z12 is a ring under addition and multiplication modulo 12.We have to find its all prime ideals…

Q: In a commutative ring of characteristic 2, prove that the idempotentsform a subring.

A: Given: Commutative ring of characteristic 2.We need to prove that the idempotents form a subring.

Q: If Ris a commutative ring, then (b) a | b and b|c⇒a | ci.e., this relation of divisibil it y in Ris…

A: Given that if  R is a commutative ring, then we have to show that  (i) a|b and b|c then a| c i.e.,…

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: Suppose R is a commutative ring with 1_R. Let P be a prime ideal of R, I be the principal ideal…

A: Let's analyze each statement and determine whether it is true or false:(i) P = aP: This statement is…

Q: Let R be a ring without any nonzero nilpotent elements. Show that (ara - ra)² = 0 for all r E R and…

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Q: Give an example of a ring homomorphism from a commutative ringR to a ring S that maps a zero-divisor…

A: Given: Commutative ring R and S.We need to provide an example of a ring homomorphism from R→S that…

Q: IfE isa ring of even inte gers, then show that an ideal (4), generated by 4 is a maximal ideal of E.

A: If E is a ring of even integers, then our claim that an ideal (4), generated by 4 is a  maximal…

Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

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Q: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

A: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: F Let K/L be ideal of quotient ring R/L. If K is prime ideal and contains L, then K/L is prime…

A: We can solve this using defination of ideals ,prime ideals ,quotient ideals

Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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Q: Q1: what's the relation between prime and semiprime ideals.

A: As per the guidelines we are supposed yo solve the first question.kindly post another questions…

Q: Identify all distinct maximal ideals that contain the ideal ⟨x^3 + 3⟩ in Z_5[x].

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Q: Let R be a ring such that for each a belong to R there exists x belong to R such that a^2x =a. Prove…

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Q: An element x of a ring R is idempotent if x2 = x. (a) Find four idempotent elements of GL(2, R (this…

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Q: show that an ideal A is maximal iff A is

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Q: Q) Put True (T) or Fuls (F) for the following: (1)Every prime ideal I of ring R is maximal ideal.…

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Q: 1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т…

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Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

Q: Let P be a prime number. Let Zp) be the following subset of Q: a Z(p) = {₁ where a € Z, bЄ N, such…

A: Let p be a prime number and Zp be the following subset of Q. Zp=ab, where a∈Z , b∈N , such that p…

Q: An ideal A of a commutative ring R with unity is said to be finitely generated if there exist…

A: Given:A is an ideal of a commutative ring R with unity. Also, there exists a1, a2, . . . , an∈A such…

Q: 1. Let M be a proper ideal in a Boolean ring R with unity. Prove that (i) R/M is a Boolean ring and…

A: It is given that, M is a proper ideal in a Boolean ring R with unity.

Q: Let f∶ R → S be a ring homomorphism,with R and S commutative.If P is a prime ideal of S,show that…

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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: (5) In commutative ring with identity each proper ideal is contained in prime ideal. От F

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Q: III. An element x of a ring R is called idempotent if x² = x. Determine all the idempotent elements…

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