Free Module with a Projective Sub- Module, and Tensor Products - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:14:38Z http://mathoverflow.net/feeds/question/100136 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100136/free-module-with-a-projective-sub-module-and-tensor-products Free Module with a Projective Sub- Module, and Tensor Products Milan Bernolak 2012-06-20T13:59:43Z 2012-06-20T23:12:07Z <p>Let us consider a unital algebra $A$, with a subalgebra $B \subseteq A$, along with an $A$-$A$-bimodule $M$ which is free as a right module, and a subspace $N$ (with respect to the action of the field coming from the unit of $A$) such that $BNB \subseteq N$, and $N$ is a right $B$-projective module.</p> <p>The tensor product $M \otimes_A M$ is of course again a right free $A$-$A$-bimodule, and the tensor product $N \otimes_B N$ is again projective as a right $B$-module. What I would like to know is whether the canonical insertion of $N \otimes_B N$ into $M \otimes_A M$ is an embedding?</p> http://mathoverflow.net/questions/100136/free-module-with-a-projective-sub-module-and-tensor-products/100190#100190 Answer by Jeremy Rickard for Free Module with a Projective Sub- Module, and Tensor Products Jeremy Rickard 2012-06-20T23:12:07Z 2012-06-20T23:12:07Z <p>No. Take $N=M=A$, where $A$ is any non-trivial algebra over a field $k$, and $B=k$.</p>