7. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is not prime ideal then..... (a) 3a, b e R:a.beI = a € and b € I (b) Va, be R: a. beI > a ¢ I and b e I (c) va, be R:a.beI = a € l or bel (d) No Choice (a) (b) (c) (d) O 8. In the field (Z,, +7,.7), the set of all prime elements is... (a) p (b) (2,3,5} (c) Z, (d) No Choice (a) (b) (c) (d)

Q: 3. Let R. be a commutative ring with ideals L and M. Give an example of when LMLn M.

A:

Q: Q:- Consider a ving R={o,a,b,c,d, e,f.g} defired by folowing the cayley's tables. elf ef +) a e a d…

A: Solution: The given ring is R=0,a,b,c,d,e,f,g We have to calculate the characteristic of the ring R.

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

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

Q: your answer fully. (i) If I and J are prime ideals in R then In J is a prime ideal in R. (ii) If I…

A:

Q: 5. Let R be a ring and r1,., r, € R. Prove that the subset (r1,., "n) = {Airı + Anrn | A1,. An E R}…

A:

Q: 12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and…

A: Given that R is a ring without zero divisors. If ab=1, show that ba=1

Q: Theorem 79. Let a, b, c, d ∈ R where R is a ring then (a + b)(c + d) = ac + ad + bc + bd. Note.…

A:

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: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

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

A:

Q: 18.1.2. Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a…

A: To show this, let's consider a counterexample. Let R be the ring of integers Z, and let a = 2 and b…

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: If R and S are rings, define the direct sum of R and S, ROS by RS = {(r, s) : r € R, s € S} where…

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: 3. In a Ring (R, +, *) let '0' be identity of '+' and '1' be identity of only one element. unique…

A:

Q: Hom worke: Consider the ring (Z [√3], +,.), Let A={ a+b√3:a, be Z₂} Is A subring of Z[√3]?

A:

Q: Let a ER which expressions are part of the Borel o algebra, B(R) they are correct? and give the…

A: The main objective is to determine the correct option.

Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

A:

Q: 7. Aring with identity (R. +..) is commutative if: (a) Ris an integral domain () (a+ b) - a + Zab +…

A:

Q: 1. If (R, +,.) be a ring without divisors of zero then: (a) R has 1 (b) the cancellation law holds…

A:

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

A:

Q: Theorem 3. Let R be a ring. Then for all a,b,c E R, 1. a0 =0= 0a. 2. a(-b) = -(ab) = (-a)b. 3.…

A:

Q: Demonstrate the immediate properties of an A ring 1) If a ∈ A, then - (- a) = a 2) Data x, y, z ∈…

A: Definition: If R is a ring and a∈R, then -a is a unique solution of a+x=0. 1) Given that A is a ring…