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If the least common multiple of a and b in the Euclidean ring R is denoted by [a,b], prove that [a,b] = ab / (a,b) , where (a,b) is the greatest common divisor.
Let p be a prime and let G be an abelian group of order a power of p. Prove that the nilradical of the group ring G is the augmentation ideal.
Let R be a ring such that the only right ideals of R are (0) and R. Prove that either R is a division ring or, that R is a ring.
Prove that I = is not a prime ideal of Z[i]. How many elements are in Z[i] / I. What is the characteristic of Z[i] / I.
Let F be the set of all functions f : R ? R. We know that is an abelian group under the usual function addition, (f + g)(x) = f(x) + g(x).
Let F, K be two fields F ? K and suppose f(x), g(x) ? F[x] are relatively prime in F[x]. Prove that they are relatively prime in K[x].
Let R be a ring with 1 and let S=M2(R). If I is an field of S .show that there is an ideal J of R such that I consists of all 2X2 matrices over J.
Prove that x2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x] / (x2 + 1) is a field having 121 elements.
If a is rational and x - a divides an integer monic polynomial, prove that a must be an integer.
Let F be a field of real numbers. Prove that F[x] / (x2 + 1) is a field isomorphic to the field of complex numbers.
Let D be a Euclidean ring, F its field of quotients. Prove the Gauss lemma for polynomials with coefficients in D factored as product of polynomials.
If P is a prime number, prove that the polynomial xn - p is irreducible over the rationals.
A ring of sets is a non-empty class A of sets such that if A and B are in A, then A?B and AnB are also in A. Show that A must also contain A - B.
Show that if R is a ring with identity element and x is an element in R, then Rx = {rx: r in R} is the principal left ideal generated by x.
If R is an integral domain, prove that for any two non-zero elements f(x) , g(x) of R[x], deg(f(x)g(x)) = deg f(x) + deg g(x)
Prove that if R is a commutative ring with unit element then R[x] is also a commutative ring with unit element.
If R is an integral domain with unit element, prove that any unit in R[x] is a unit in R and any unit in R is also a unit in R[x].
Let R be a commutative ring with no non-zero nilpotent elements ( that is, an = 0 implies).
If R is a unique factorization domain and if a, b are in R, then a and b have a least common multiple (l.c.m.) in R.
Prove that if R is an integral domain, then R[x1 , x2, .....,xn] is also an integral domain.
Let R={f(x) included in Q[x]: the coefficient of x in f(x) is 0} Prove that R is a subring of Q[x].
If the sequence is split over S, then it is split over R. If the sequence is split over R, then it is split over S.
Show that the class of all finite unions of sets of the form A X B with A ? A and B ? B is a ring of subsets of X ? Y .
Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of O containing the identity.
Prove conversely that if R is a commutative ring with 1 in which the set of non-units forms an ideal M, then R is a local ring with unique maximal ideal M.