Show that the intersection of any family of ideals in a ring is an ideal.
Show that the ideal generated by a subset S of a ring R is the intersection of all ideals J of R such that S <= J <=R.

Let R be a ring such that the only right ideals of R are (0) and R.
Prove that either R is a division ring
or, that R is a ring with a prime number of elements in which ab = 0 for
every a,b Є R.

A commutative ring is called a local ring if it has a unique maximal ideal. Prove that if R is a local ring with maximal ideal M then every element of R-M is a unit. Prove conversely that if R is commutative ring with one in which the set of nonunits forms an ideal M, then R is a local ring with unique maximal ideal M.

A commutative ring R is called a local ring if it has a unique maximal ideal. Prove that if R is a local ring with maximal ideal M then every element of (R-M) is a unit.
Prove conversely that if R is a commutative ring with 1 in which the set of non-units forms an ideal M, then R is a local ring with unique maximal ideal M.

If n Є R and R is a commutative ring we indicate by Mn(R) the ring of allnxn entries wrt the usual operations on matrices. If n>1 this ring is commutative even if R is.
Let S={(aij)ЄMn(R)|i≠j=>aij=0}
Let k be an integer 1≤k≤n. Show that
a) S is a commutative subring of Mn(R)
b) The function f: S

With respect to the ideal
I=<2,x> in Z[x]
I believe this ideal is maximal because one theorem I have read suggests to me that all maximal ideals of Z[x] are in the form

where p is prime and f(x) is an element of Z[x] and irreducible mod p. It appears that <2,x> fits this description.
Did I understand correc

(a) The kernel of this homomorphism is the principal ideal (x-1). Therefore, Z[x]/(x-1) is isomorphic to Z. According to the correspondence theorem, ideals of Z[x]/(x-1) are in one-to-one correspondence with ideals of Z[x] containing (x-1). Taking into account the above-mentioned isomorphism, we obtain that ideals of Z are in

Let delta=sqrt(-3) and R=Z[delta]. This is not the ring of integers in the imaginary quadratic number field Q[delta]. Let A be the ideal (2,1+delta).
a) Show that A is a maximal ideal and identify the quotient ring R/A.
b) Show that A contains the principal ideal (2) but that A does not divide (2).

Please help with the following problems.
a) If I,J are ideals in a ring R such that I+J=R and R is isomorphic to the product ring (R/I)x(R/J) when IJ=0, describe the idempotents corresponding to this product decomposition;
b) Describe the maximal ideal of RxR where in this case R is the set of real numbers;
c) How many roo

1.Give a example of a commutative ring that has a maximal ideal that is not a prime ideal.
2. Prove that I=<2+2i> is not a prime ideal of Z[i]. How many elements are in Z[i]/I ? What is the characteristic of Z[i]/I ?
Please see the attached file for the fully formatted problems.