Abstract Algebra Answers

For a triangular ring T =
R M
0 S
(where M is an (R, S) - bimodule), show that rad(T) =
rad(R) M
0 rad(S)

Show that, for any direct product of rings Ri, rad ((direct product)Ri) = (direct product) rad Ri.

Let Ai (i ∈ I) be ideals in a ring R, and let A =(intersection on i) Ai. True or False: “If each R/Ai is J-semisimple, then so is R/A”?

Show that rad R is the smallest ideal I ⊆ R such that R/I is J-semisimple.

If an ideal I ⊆ R is such that R/I is J-semisimple, show that I ⊇ rad R.

Show that RM is simple iff M ∼ R/m (as left R-modules) for a suitable modular maximal left ideal m ⊂ R.

Show that rad R is the intersection of all modular maximal left (resp. right) ideals of R.

Define the Jacobson radical of R by rad R = {a ∈ R : Ra is left quasi-regular}.
Show that, if R has an identity, the definition of rad R here agrees with classical one .

Show that rad R is a quasi-regular ideal which contains every quasi-regular left (resp. right) ideal of R. (In particular, rad R contains every nil left or right ideal of R.)

Show that if a left ideal I ⊆ R is left quasi-regular, then it is quasi-regular.