Question

ar URSCH. In act prove that the identity map is the only ring isomorphism of 2. Let a and b be nonzero elements of the Unique Factorization Domain R. Prove that a and b have a least common multiple (cf. Exercise 11 of Section 1) and describe it in terms of the prime factorizations of a and b in the same fashion that Proposition 13 describes their greatest common divisor. 3. Determine all the representations of the integer 2130797 = 172.73.101 as a sum of two

Answer 1

Recall by definition, a least common multiple of a and b is an element e of R(commutative ring with 1) such that a|e and b|e; and  if a|e′ and b|e′ then e|e′.

This least common multiple of a and b (if such exists) is a generator for the unique largest principal ideal contained in (a)∩(b).

Recall that every principal ideal domain is a unique factorization domain.

And we know that, any two non-zero elements of a P.I.D. have a least common multiple.

Since, P.I.D implies U.D.F .

So, any two non-zero elements of a P.I.D. have a least common multiple implies that a and b have a least common multiple.

For the prime factorisation domain,

let a and b be two nonzero elements of the Unique Factorization Domain Rand suppose a = upp?.... pand b = vp-p2.... po tome are prime factorizations for a and b, where u and v are units, the primes P1, P2, ...., Pnare distinct and the exponents e; and fi are > 0. Then the element 1=prar(e1,1) pmax(e2,82).... pmaz(em,fa) is a least common multiple of a and b.

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