Homework 1

Problem 1

Suppose R is a ring, M is a left R-module, and rR is an element for which there exists a nonzero mM such that rm=0. Prove r does not have a left inverse.


Problem 2

Suppose R is a ring, IR is a left ideal, and M is a left R-module. Let IMM denote the subset of all finite I-linear combinations in M, i.e., IM={finiteikmkikI,mkM}. Prove IM is a submodule of M.


Problem 3

Suppose R is a ring and M is a left R-module. An element mM is called a torsion element if rm=0 for some nonzero rR. The set of torsion elements in M is denoted Tor(M).

  1. Prove that if R is an integral domain then Tor(M) is a submodule of M.
  2. Give an example of a ring R and R-module M such that Tor(M) is not a submodule. (Hint: Consider torsion elements in the R-module R for some specific ring R.)
  3. Suppose R has a (nonzero) zero divisor and M is nontrivial. Prove that M has nonzero torsion elements.
  4. Suppose ϕ:MN is an R-module morphism. Prove that ϕ(Tor(M))Tor(N).

Problem 4

Suppose R is a ring and M is a left R-module.

  1. For each submodule N on M, the annihilator of N in R is defined to be the set of elements rR such that rn=0 for all nN. Prove that the annihilator of N in R is a 2-sided ideal of R.
  2. For each right ideal I of R, the annihilator of I in M is defined to be the set of all elements mM such that im=0 for all iI. Prove that the annihilator of I in M is a submodule of M.
  3. Consider the Z-module M=Z24×Z15×Z50 and ideal I=2Z. Determine the annihilator of M in Z and the annihilator of I in M.

Problem 5

Give an example of a ring R, two R-modules M and N, and a set map f:MN such that f is a group morphism but not an R-module morphism.


Problem 6

Suppose R is a commutative ring and M is left R-module. Prove that HomR(R,M) and M are isomorphic as left R-modules.

Bonus challenge: Is your isomorphism natural in M?


Problem 7

Suppose R is a commutative ring. Prove that HomR(R,R) and R are isomorphic as rings.