1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т P.Q := {Ea; • b; | m e N, a; e P, b; e Q }, i=1 and the sum is defined as P+ Q := {a+b|a E P , be Q } . P +Q := (i) Show that P·Q, P+Q, and PNQ are ideals in R, and that P·Q C PNQ. (ii) Show that if P + Q and both are maximal ideals then P· Q = PnQ.

Q: 7. Find all distinct principal ideals of Z14

A: Since Z14 are cyclic group with respect to addition so all ideals of Z14 are principal ideals

Q: 2 Consider the bases B₁ and B₂. 2 -3 -7 4 {{³][])} {₁ 6 -5 B₁ = Find [x]B₂ if [x]B₁ 2 [²]. 6

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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: (Q1)Find all idempotent elements and nilpotent elements of the following rings 2- (Z/(3),+..) + (C.…

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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: Use the inner product = aobo+ aıb1+ azbz+ asb; to find , lIlp|], d(p, q), and then find the angle…

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Q: Que 405: Let S be a prime ideal of R and suppose that a product 1a2a3... an, of elements of R,…

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Q: One of the following is an ideal of z1: (0,3} (0,2,4,6} None (9,3,6,9}

A: From given statement

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Q: FOR ANY TWOCOMPLEX NUMBERSZ, ANDZ,,PROVE THAT |Z|+|Z,|=,(Z, +Z,)+ /Z,Z» +÷(Z, +Z»)- Z,Z, .

A: for two complex number Z1 and Z2 prove that Z1+Z2=12Z1+Z2+Z1Z2+12Z1+Z2-Z1Z2

Q: Find gcd (272, 1479) and find the integers x, y such that gcd(272, 1479) = 272x + 1479y

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Q: THE RADICAL OF AN IDEAL IS ALSO AN IDEAL.

A: We have to show that  RADICAL OF AN IDEAL IS AN IDEAL

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Q: 2. Determine whether or not the set is an ideal in the ring M2×2(R). J-{(8₂9) | PCR} = ER

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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

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Q: 3. npor Find the dot product

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

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Q: 9. The number of ideals in the ring (Zis.+1s1s) is: (a) Two (b) Three (e) Four (d) Five.

A: The positive divisor of 15 are: 1,3,5,15

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: Q3) Use the inner product = a,bo + a,b1 + azb2 to find , llp||, d (p, q), and then find the angle…

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Q: Let A = (a1, . . as) and B = defines the product of the ideals to be (B1,.. Bt) be ideals in K. The…

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