let R be a commutative ring with 1 and let a, beR-307. The L.C.M of a,b...

Transcribed Image Text:

let R be a commutative ring with 1 and let a, beR-307. The L.C.M of a,b is an element eeR such that: (1) ale, ble and (ii) if alf and blf, then elf. (a) Prove that a lcm of a,b (if it exists) is a generator of the only principal ideal, which is the largest (regarding contention), Contained in (a) n<b> (b) Inter that in an Integral Domain, the Icm of a,b is unique (up to Unities) one (c) Prove that in an Euclidean Domain, (cm(a, b)= (d) find a in Q[x]. ab MCD (a,b) generator for (x²-1, x²-1) and a generator for (x²-1)/(x²-1) let R be a commutative ring with 1 and let a, beR-307. The L.C.M of a,b is an element eeR such that: (1) ale, ble and (ii) if alf and blf, then elf. (a) Prove that a lcm of a,b (if it exists) is a generator of the only principal ideal, which is the largest (regarding contention), Contained in (a) n<b> (b) Inter that in an Integral Domain, the Icm of a,b is unique (up to Unities) one (c) Prove that in an Euclidean Domain, (cm(a, b)= (d) find a in Q[x]. ab MCD (a,b) generator for (x²-1, x²-1) and a generator for (x²-1)/(x²-1)