Abstract Algebra Answers

Is any subring of a left semisimple ring left semisimple?

Let R be a ring with center C. Show that a right ideal A of R is an ideal if:
the factor group R/A is cyclic, or isomorphic to a subgroup of Q.

Let R be a ring with center C. Show that a right ideal A of R is an ideal if:
R/A is a cyclic left C-module.

Let R be a ring with center C. Show that a right ideal A of R is an ideal if: End(C(R/A)) is a commutative ring;

S = IR(A) be the idealizer of A. Show that (1), (2) are equivalent:
(1) A is a maximal right ideal and R/A is a cyclic left S-module.
(2) A is an ideal of R, and R/A is a division ring.

S = IR(A) be the idealizer of A.
Show that (1), (2) are equivalent:
(1) End(S(R/A)) is a commutative ring.
(2) A is an ideal of R, and R/A is a commutative ring.

the idealizer IR(A) and the eigenring ER(A) for the right ideal A = xR in the free k-ring R =
k<x, y>, and for the right ideal A = xR where x = i + j + k in the ring of quaternions with integer coefficients.

Let R = Mn(k) where k is a ring, and let A be the right ideal of R consisting of matrices whose first r rows are zero. Compute the idealizer IR(A) and the eigenring ER(A).

i have 51 skittle 15 yellow 6 red 13purple orange 17and green 12 i have to write the ratio of the number of skittles to mass of packet (grams)