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Prove that if M is a maximal ideal of S and phi is surjective then phi^-1(M) is maximal ideal of R.
Show that the field of fractions of the power series ring Z[[x]] is properly contained in the field of Laurent series Q((x)).
Let m and n be positive integers with n dividing m. Prove that the natural surjective ring projection Z/mZ ->Z/nZ is also surjective on the units.
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.