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. (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.

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A: As this question has multiple subparts, hence only first five subparts will be done.

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