# Rings.

• Apr 23rd 2010, 11:53 AM
tom007
Rings.
1.Let p_1,p_2 e Z[x]. Z[p_1,p_2] is subring of Z[x] generated with Z U {p_1,p_2}
are Z[x^2 - x^5, x^2 - 2x^5], Z[x^2+x^6,x^2+2x^6] unique factorization domain?

2.Prove that the ring Z[2 *sqrt(-1)]={a+2b* sqrt(-1), a,b e Z} is not principial ideal domain. Is it Euclidean domain?

• Apr 23rd 2010, 05:48 PM
Drexel28
Quote:

Originally Posted by tom007
1.Let p_1,p_2 e Z[x]. Z[p_1,p_2] is subring of Z[x] generated with Z U {p_1,p_2}
are Z[x^2 - x^5, x^2 - 2x^5], Z[x^2+x^6,x^2+2x^6] unique factorization domain?

2.Prove that the ring Z[2 *sqrt(-1)]={a+2b* sqrt(-1), a,b e Z} is not principial ideal domain. Is it Euclidean domain?

I mean, you've shown zero work. We have no idea where you are at in your skill levels?

What does generate mean?

What is a unique factorization domain (UFD)?

What is a principle ideal domain (PID)?

What is a Euclidean Domain? (or maybe what is a Euclidean valuation?)
• Apr 23rd 2010, 11:54 PM
tom007
An integral domain R is a unique factorization domain provided that:
(i) every nonzero nonunit element a of K can he written a = C1C2...Cn, with
C1, . . . , Cn irreducible.
(ii) if a = C1C2.... Cn and a = d1d2....dm (Ci,di irreducible), then n = m and for
some permutation o of { 1,2, . . . , n}, Ci and d(o(i)) are associates for every i.

An ideal (x) generated by a single element is called a principal ideal. A principal ideal ring is a ring in which every ideal is principal. A principal ideal ring which is an integral domain is called a principal ideal domain.

R is a Euclidean ring if there is a function v : R — {0) -> N such that:
(i) if a,b e R and ab=! 0, then v(a) <=v(ab);
(ii) if a,b e R and b=! 0, then there exist q,r s R such that a = qb + r with r= 0, or r=!0 and v(r) < v(b).
A Euclidean ring which is an integral domain is called a Euclidean domain.