1. Introduction
We consider only nonzero associative unital rings. For a ring
R, we write w.gl.dim.
if
R is a
ring of weak global dimension at most one, i.e.,
R satisfies the following equivalent (The equivalence of the conditions is well known; e.g., see the conditions in [
1] (Theorem 6.12)).
For every finitely generated right ideal X of R and each finitely generated left ideal Y of R, the natural group homomorphism is an isomorphism.
Every finitely generated right (resp., left) ideal of R is a flat right (resp., left) R-module.
Every right (resp., left) ideal of R is a flat right (resp., left) R-module.
Every submodule of any flat right (resp., left) R-module is flat.
for all right (resp., left) R-modules A and B.
Since every projective module is flat, any right or left (semi)hereditary ring is of weak global dimension at most one. (a module
M is said to be hereditary (resp., semihereditary) if all submodules (resp., finitely generated submodules) of
M are projective.) We also recall that a ring
R is of weak global dimension zero if and only if
R is a Von Neumann regular ring, i.e.,
for every element
r of
R. Von Neumann regular rings are widely used in mathematics; see [
2,
3].
A ring R is said to be arithmetical if the lattice of two-sided ideals of R is distributive, i.e., for any three ideals of R. A ring R is said to be semiprime (resp., prime) if R does not have nilpotent nonzero ideals (resp., the product of any two nonzero ideals of R are nonzero).
Theorem 1. (C.U.Jensen ([
4], Theorem))
. A commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring
R with center
C is said to be
centrally essential if
is an essential extension of the module
, i.e., for every nonzero element
, there exist two nonzero central elements
with
. Centrally essential rings are studied in many papers; e.g., see [
5].
There are many noncommutative centrally essential rings. For example, if
F is the field
and
is the quaternion group of order 8, then the group algebra
is a finite noncommutative centrally essential ring; see [
5].
Let F be the field , and let V be a vector F-space with basis . It is known that the exterior algebra of the space V is a finite centrally essential noncommutative ring. It is known that there exists a centrally essential ring R such that the factor ring with respect to the Jacobson radical is not a PI ring (in particular, the ring is not commutative).
A module M is said to be distributive (resp., uniserial) if the submodule lattice of M is distributive (resp., is a chain). It is clear that a commutative ring is right (resp., left) distributive if and only if the ring is arithmetical.
The main result of this work is Theorem 2.
Theorem 2. For a centrally essential ring R, the following conditions are equivalent.
- 1.
R is a ring of weak global dimension at most one.
- 2.
R is a right (resp., left) distributive semiprime ring.
- 3.
R is an arithmetical semiprime ring
2. Remarks and Proof of Theorem 2
Example 1. The implication (1) ⇒ (2) of Theorem 2 is not true for arbitrary rings. There exists a right hereditary ring R of weak global dimension at most one that is neither right distributive nor semiprime; in particular, the right hereditary ring R is of weak global dimension at most one. Let F be a field, and let R be the 5-dimensional F-algebra consisting of all matrices of the following form: , where . The ring R is not semiprime, since the following set is a nonzero nilpotent ideal of R: . Let , , and be ordinary matrix units. The ring R is not right or left distributive, since every idempotent of a right or left distributive ring is central (see [6]), but the matrix unit of R is not central. To prove that the ring R is right hereditary, it is sufficient to prove that is a direct sum of hereditary right ideals. We have that , where and are projective simple R-modules; in particular, and are hereditary R-modules. Any direct sum of hereditary modules is hereditary; see ([7], 39.7, p. 332). Therefore, it remains to show that the R-module is hereditary, which is directly verified. The following lemma is well known; e.g., see ([
1], Assertion 6.13).
Lemma 1. Let R be a ring in which the principal right ideals are flat. If r and s are two elements of R with , then there exist two elements such that , , and .
Lemma 2. Let R be a centrally essential ring in which the principal right ideals are flat. Then, the ring R does not have nonzero nilpotent elements.
Proof. Indeed, let us assume that there exists a nonzero element with . Since the ring R is centrally essential, there exist two nonzero central elements with . Since , we have that . Since , it follows from Lemma 1 that there exist two elements such that , , and . Then, . This is a contradiction. □
Lemma 3. There exists right and left uniserial prime rings R that habe a non-flat principal right ideal.
Proof. There exists right and left uniserial prime rings
R with two nonzero elements
such that
; see ([
8], p. 234, Corollary). The uniserial ring
R is local; therefore, the invertible elements of
R form the Jacobson radical
of
R. The ring
R is not a ring in which the principal right ideals are flat. Indeed, let us assume the contrary. By Lemma 1, there exist two elements
such that
,
, and
. We have that either
or
; in addition,
. Therefore, at least one of the elements
of the local ring
R is invertible; in particular, this invertible element is not a right or left zero-divisor. This contradicts to the relations
and
. □
Remark 1. It follows from Lemma 3 that the implication (2) ⇒ (1) of Theorem 2 is not true for arbitrary rings.
Lemma 4. Every centrally essential semiprime ring R is commutative.
Proof. Assume the contrary. Then, the ring
R does not coincide with its center
C and
for some
. We note that
is an ideal of the ring
C. The set
is not empty, since we can take
. We take any element
with
. If
, then
for some
. Hence
, and therefore,
and
. Thus,
and
; this is a contradiction. Therefore,
, and thus,
. Therefore,
and
. This implies that
. For any
, we have that
. Thus,
and
. However,
for some nonzero elements
, so
and, hence,
; this is a contradiction. Thus,
R is commutative. □
The Completion of the Proof of Theorem 2
Proof. (1) ⇒ (2). Since R is a centrally essential ring of weak global dimension at most one, it follows from Lemma 2 that the ring R does not have nonzero nilpotent elements. By Lemma 4, the centrally essential semiprime ring R is commutative. By Theorem 1, R is an arithmetical semiprime ring. Any commutative arithmetical ring is right and left distributive.
The implication (2) ⇒ (3) follows from the property that every right or left distributive ring is arithmetical.
(3) ⇒ (1). Since R is a centrally essential semiprime ring, it follows from Lemma 4 that the ring R is commutative; in particular, R is centrally essential. In addition, R is arithmetical. By Theorem 1, the ring R is of weak global dimension at most one. □