- 11.1 Prime and primitive ideals
- 11.2 The Jacobson radical
- 11.3 Semisimple Artinian rings

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A proper ideal I of the ring R is called a

A proper ideal P of the ring R is called a

**11.1.2. Definition.**
The ring R is called a
** simple ring **
if (0) is a maximal ideal;
it is called a
** prime ring **
if (0) is a prime ideal, and a
** semiprime ring **
if (0) is a semiprime ideal.
Finally, R is said to be a (left)
** primitive ring **
if (0) is a primitive ideal.

**11.1.3. Proposition.**
The following conditions are equivalent
for the proper ideal P of the ring R:

**(1)**
P is a prime ideal;

**(2)**
AB P implies
A P or
B P,
for any ideals A, B of R which contain P;

**(3)**
AB P implies A = P or B = P,
for any left ideals A, B of R with
P A and
P B;

**(4)**
aRb P implies
a P or
b P,
for any a,b R.

**11.1.4. Definition.**
The nonzero module
_{R}M is called a
** prime module **
if AN = (0) implies N = (0) or AM = (0),
for any ideal A of R and any submodule N of M.

**11.1.5. Proposition.**
The annihilator of a prime module is a prime ideal.

**11.1.6. Proposition.**
Any maximal ideal is primitive,
and any primitive ideal is prime.

In a left
Artinian
ring, the notions of maximal ideal, primitive ideal, and prime ideal coincide.

An ideal I of the ring R is said to be
** nilpotent **
if A^{n} = (0)
for some positive integer n.
It is said to be a
** nil ideal **
if for each
a I
there exists a positive integer n such that
a^{n} = 0.

**11.1.7. Proposition.**
The following conditions are equivalent
for the proper ideal I of the ring R:

**(1)**
I is a semiprime ideal;

**(2)**
the ring R/I has no nonzero nilpotent ideals;

**(3)**
AB I implies
AB
I,
for any ideals A, B of R;

**(4)**
AB I implies
AB = I,
for any left ideals A, B of R
with I A and
I B;

**(5)**
aRa I implies
a I,
for all a R.

**11.1.8. Definition.**
Let
_{R}M be a left R-module. The
** bicommutator **
of M is the subring of
End_{Z}(M)
defined by

Bic_{R}(M) =
{
End_{Z}(M) | f
=
f for all f
End_{R}(M) }.

**11.1.10. Proposition.**
If P is a primitive ideal of the ring R,
then there exists a
division ring
D and a vector space V over D for which R/P
is isomorphic to a subring of the ring
of all linear transformations from V into V.

**11.1.11. Proposition.**
Let M be a left R-module.

**(a)**
The endomorphism ring
End_{R}(M^{n})
is isomorphic to the ring of
n×n
matrices with entries in
End_{R}(M).

**(b)**
The bicommutator
Bic_{R}(M^{n})
is isomorphic to
Bic_{R}(M).

**11.2.2. Definition.**
Let M be a left R-module.

The submodule N of M is called
** essential ** or ** large **
in M if
NK
(0)
for all nonzero submodules K of M.

The submodule N is called
** superfluous ** or ** small **
in M if N+K M
for all proper submodules K of M.

**11.2.3. Proposition.**
Let N be a submodule of
_{R}M.
If K is maximal in the set of all submodules of M
that have trivial intersection with N,
then N+K is essential in M,
and (N+K)/K is essential in M/K.

**11.2.4. Proposition.**
The socle
of any module is the intersection of its essential submodules.

**11.2.5. Definition.**
A
** radical **
for the class of left R-modules is a function that assigns to each module
_{R}M a submodule
(M) such that

**(i)**
f((M))
(N),
for all modules
_{R}N and all
f Hom_{R}(M,N);

**(ii)**
(M/(M)) = (0).

**11.2.6. Definition.**
Let C be any class of left R-modules. For any module
_{R}M
we make the following definition.

rad_{C}(M) =
ker(f),

where the intersection is taken over all R-homomorphisms
f : M -> X, for all X in C.

**11.2.7. Proposition.**
Let
be a radical for the class of left R-modules,
and let F be the class of left R-modules X for which
(X) = (0).

**(a)**
(R) is a two-sided ideal of R.

**(b)**
(R) M
(M)
for all modules
_{R}M.

**(c)**
rad_{F}
is a radical, and
= rad_{F}.

**(d)**
(R) =
Ann(X),
where the intersection is taken over all modules
X in F.

**11.2.8. Lemma.
[Nakayama]
**
If
_{R}
M
is finitely generated and J(R)M = M, then M = (0).

**11.2.9. Proposition.**
Let M be a left R-module.

**(a)**
J(M) = { m M |
Rm is small in M }.

**(b)**
J(M) is the sum of all small submodules of M.

**(c)**
If M is finitely generated, then J(M) is a small submodule.

**(d)**
If M is finitely generated, then M/J(M) is
semisimple
if and only if it is Artinian.

**11.2.10. Theorem.**
The Jacobson radical J(R) of the ring R
is equal to each of the following sets:

**(1)**
The intersection of all maximal left ideals of R;

**(2)**
The intersection of all maximal right ideals of R;

**(3)**
The intersection of all left-primitive ideals of R;

**(4)**
The intersection of all right-primitive ideals of R;

**(5)**
{ x R |
1-ax is left invertible for all
a R };

**(6)**
{ x R |
1-xa is right invertible for all
a R };

**(7)**
The largest ideal J of R such that 1-x is invertible in R for all
x J.

**11.2.11. Definition.**
The ring R is said to be
** semiprimitive **
if J(R) = (0).

**11.2.12. Proposition.**
Let R be any ring.

**(a)**
The Jacobson radical of R contains every nil ideal of R.

**(b)**
If R is left Artinian, then the Jacobson radical of R is nilpotent.

**11.3.2. Theorem.
[Artin-Wedderburn]
**
The following conditions are equivalent for a ring R with identity.

**(1)**
R is left Artinian and J(R) = (0);

**(2)**
_{R}R is a semisimple module;

**(3)**
R is isomorphic to a finite direct product of rings of
n×n matrices over division rings.

**11.3.3. Definition.**
A ring which satisfies the conditions of
the previous theorem is said to be
** semisimple Artinian**.

**11.3.4. Corollary.**
The following conditions are equivalent for a ring R with identity.

**(1)**
R is semisimple Artinian;

**(2)**
R is left Artinian and semiprime;

**(3)**
every left R-module is
completely reducible;

**(4)**
every left R-module is
projective;

**(5)**
every left R-module is
injective.

**11.3.5. Theorem.
[Hopkins]
**
Any left Artinian ring is left Noetherian.

**11.3.6. Lemma.**
Let _{R}M be a simple module, and let
D = End_{R}(M).
If _{D}V
is any finite dimensional subspace of
_{D}M, then

V = { m M | Ann_{R}(V) m = 0 }.

**11.3.7. Theorem.
[Jacobson Density Theorem]
**
Any (left) primitive ring is isomorphic to
a dense ring of linear transformations
of a vector space over a division ring.

**11.3.8. Proposition.**
If _{R}M
is completely reducible and finitely generated, then
End_{R}(M)
is isomorphic to a finite direct product of rings of
n×n matrices over division rings.

**11.3.9. Definition.**
A ring R is called
** von Neumann regular **
if for each
a R there exists
b R such that aba = a.

**11.3.10. Proposition.**
If _{R}M
is completely reducible,
then End_{R}(M)
is von Neumann regular.

**11.3.11. Proposition.**
The following conditions are equivalent for a ring R.

**(1)**
R is von Neumann regular;

**(2)**
each
principal
left ideal of R is generated by an
idempotent
element;

**(3)**
each finitely generated left ideal of R
is generated by an idempotent element.

**11.1.12. Proposition.**
If R is a von Neumann regular ring, then J(R) = (0).