Tensor Products of Modules

Let \(I\) be the ideal \((2,1+\sqrt {-5})\) in the ring \(R := \mathbb {Z}[\sqrt {-5}]\). Prove that \(I\otimes _R I\) is isomorphic to \(2R\).

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Consider the maximal ideal \(\langle x,y\rangle \subset k[x,y]\) for a field \(k\).

1. Prove that the kernel of the natural map of \(k[x, y]\)-modules:

   \[ \langle x,y\rangle \otimes _{k[x,y]}\langle x,y\rangle \to \langle x,y\rangle ^2 \]

   is a one-dimensional vector space over \(k\) and find a generator of it.

2. Find the other two Tors:

   \[ \operatorname {Tor}^{k[x,y]}_1(\langle x,y\rangle ,\langle x,y\rangle )\quad \text {and}\quad \operatorname {Tor}^{k[x,y]}_2(\langle x,y\rangle ,\langle x,y\rangle ) \]

Let \(k\) be the field with 2 elements and let \(R=k[x]\). Identify, up to isomorphism, all \(R\)-modules \(M\) with 8 elements (that is \(|M|=8\)) and such that \(M\otimes _R (R/(x^2(x+1)))\) also has 8 elements.

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Let \(d\) be the greatest common divisor of positive integers \(n\) and \(m\). Prove that

Let \(F\) be a field and \(n\) a positive integer such that \(F\) has no nontrivial field extensions of degree less than \(n\). Let \(L=F[\alpha ]\) be an extension such that \(\alpha ^n\) is in \(F\). Prove that each element of \(L\) is a product of elements of the form \(a\alpha +b\), with \(a,b\) in \(F\).

Suppose \(A\) is a finitely generated abelian group such that \(A\otimes _{\mathbb {Z}}\mathbb {F}=0\) for all finite fields \(\mathbb {F}\). Show that \(A=0\).

Find all the maximal ideals of \(\mathbb {Z}[\sqrt {-13}]\) that contain 7, and prove that their product is precisely the ideal \((7)\).

Let \(M\) and \(N\) be finitely generated modules over a commutative local ring \(A\). If \(M\otimes _A N=0\), prove that \(M=0\) or \(N=0\). Is this true if \(A\) is not local? Prove or disprove.

If \(I,J\) are ideals in the commutative ring \(R\), prove that \[ R/I\otimes _R R/J\cong R/(I+J), \] as \(R\)-modules.

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}\cong \mathbb {Q}\otimes _{\mathbb {Q}}\mathbb {Q}\) as left \(\mathbb {Q}\)-modules.

Let \(p\) be a prime. Show that an element in the symmetric group \(S_n\) has order \(p\) if and only if it is a product of commuting \(p\)-cycles. Show by an explicit example that this need not be the case if \(p\) is not prime.

Let \(m,n\) be positive integers and \(d\) their greatest common divisor. Show that \[ \mathbb {Z}/m\mathbb {Z}\otimes _{\mathbb {Z}}\mathbb {Z}/n\mathbb {Z}\cong \mathbb {Z}/d\mathbb {Z}. \]

Show that \(1\otimes (1,1,\ldots )\ne 0\) in the tensor product \(\mathbb {Q}\otimes _{\mathbb {Z}}\prod _{n=2}^{\infty }(\mathbb {Z}/n\mathbb {Z})\).

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}=\mathbb {Q}\) and \(\mathbb {Q}\otimes _{\mathbb {Z}}(\mathbb {Z}/n\mathbb {Z})=0\).

Let \(R\) be a commutative ring with identity. Prove or disprove: the tensor product (over \(R\)) of two torsion free \(R\)-modules is torsion-free.

Let \(N\) be a normal subgroup of a finite group \(G\). Assume that the center of \(N\) is trivial and that any automorphism of \(N\) is inner. Show that there exists a normal subgroup \(H\) of \(G\) such that \(G\) is isomorphic to the direct product of \(H\) and \(N\).

Let \(R\) be a unique factorization domain with quotient field \(k\). Let \(f\) be an element of \(R[x]\). Show that if \(f\) can be factored as a product of lower-degree polynomials in \(k[x]\), then \(f\) can be factored as a product of lower-degree polynomials in \(R[x]\).

Let \(R\) be a commutative integral domain with identity. Recall that an \(R\)-module is torsion free if for any \(r\in R\) and \(m\in M\), \(rm=0\) implies that either \(r=0\) or \(m=0\).

1. Suppose \(R\) is a principal ideal domain and \(M\) and \(N\) are two \(R\) modules. Prove that if \(M\) and \(N\) are torsion free, so is \(M\otimes _R N\).
2. Give an example of an integral domain \(R\) and two torsion free \(R\)-modules \(M\) and \(N\) such that \(M\otimes _R N\) is not torsion free.

Describe the structure of a Sylow 2-subgroup of the symmetric group \(S_n\) in terms of products and semidirect products of known groups for:

1. \(n=4\); and
2. \(n=8\).

Let \(E\) be a finite separable extension of a field \(K\) and let \(F\) be any finite field extension of \(K\). Show that \(E\otimes _K F\) is a finite product of fields.

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}\cong \mathbb {Q}\), and that \(\mathbb {Q}\otimes \mathbb {Z}/N\mathbb {Z}=0\) for every positive integer \(N\), while \(\mathbb {Q}\otimes \left (\prod \mathbb {Z}/N\mathbb {Z}\right )\ne 0\), where the product is taken over all positive integers \(N\).

Let \(I\) and \(J\) be ideals of a commutative ring \(R\). Show that \((R/I)\otimes _R(R/J)\) is isomorphic to \(R/(I+J)\).

Let \(R\) be a commutative ring with an identity and assume that \(I\) and \(J\) are ideals in \(R\) such that \(I+J=R\). Prove that \(IJ=I\cap J\) and that \(R/IJ\) is isomorphic to the direct product \((R/I)\times (R/J)\).