Algebra 4 – May 15th – Map between rings

Map between rings

Let $f:A\to B$ be a morphism of rings. We define the extended ideal and the contracted ideal. Basic properties. Then to every B-module we associate an A-module via the restriction of coefficients, and to every A-module a B-module using the tensor product.

Let R be a ring, I an ideal and M a R-module. Show that $R/I\otimes_R M\cong M/IM$ both as R-modules and as R/I-modules.

If M is a free A-module, then $M\otimes_A B$ is a free B-module of same rank.

Show that $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$ is not isomorphic to $\mathbb{C}$ as vector spaces over the reals .

Definition of multiplicative system and of the ring of fractions.

References: Atiyah-Macdonald chap 1&2, link.

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