Abstract Algebra Answers

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.

Show that, if R has an identity 1, the map ϕ : (R, ◦) → (R,×) sending a to 1 − a is a monoid isomorphism. In this case, an element a is left (right) quasi-regular iff 1 − a has a left (resp. right) inverse with respect to ring multiplication.

Show that any nilpotent element is quasi-regular in every ring.

Show that if ab is left quasi-regular element of ring , then so is ba.