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

A:

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

A:

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

A:

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

A:

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…

A:

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

A:

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

A: