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Article

On Rings of Weak Global Dimension at Most One

Power Engineering Institute, National Research University, 125252 Moscow, Russia
Mathematics 2021, 9(21), 2643; https://doi.org/10.3390/math9212643
Submission received: 6 September 2021 / Revised: 14 October 2021 / Accepted: 19 October 2021 / Published: 20 October 2021

Abstract

:
A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of R is distributive. Jensen has proved earlier that 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 is said to be centrally essential if either R is commutative or, for every noncentral element x R , there exist two nonzero central elements y , z R with x y = z . In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.

1. Introduction

We consider only nonzero associative unital rings. For a ring R, we write w.gl.dim. R 1 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 X R Y X Y 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.
  • Tor 2 R ( A , B ) = 0 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., r r R r 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., X ( Y + Z ) = X Y + X Z for any three ideals X , Y , Z 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 R C is an essential extension of the module C C , i.e., for every nonzero element r R , there exist two nonzero central elements x , y R with r x = y . 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 Z / 2 Z and Q 8 is the quaternion group of order 8, then the group algebra F Q 8 is a finite noncommutative centrally essential ring; see [5].
Let F be the field Z / 3 Z , and let V be a vector F-space with basis e 1 , e 2 , e 3 . 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 R / J ( R ) with respect to the Jacobson radical is not a PI ring (in particular, the ring R / J ( R ) 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 3 × 3 matrices of the following form: f 11 f 12 f 13 0 f 22 0 0 0 f 33 , where f i j F . The ring R is not semiprime, since the following set is a nonzero nilpotent ideal of R: 0 f 12 f 13 0 0 0 0 0 0 . Let e 11 , e 22 , and e 33 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 e 11 of R is not central. To prove that the ring R is right hereditary, it is sufficient to prove that R R is a direct sum of hereditary right ideals. We have that R R = e 11 R e 22 R e 33 R , where e 22 R and e 33 R are projective simple R-modules; in particular, e 22 R and e 33 R 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 e 11 R = e 11 F + e 12 F + e 13 F 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 r s = 0 , then there exist two elements a , b R such that a + b = 1 , r a = 0 , and b s = 0 .
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 r R with r 2 = 0 . Since the ring R is centrally essential, there exist two nonzero central elements x , y R with r x = y . Since r 2 = 0 , we have that y 2 = ( r x ) 2 = r 2 x 2 = 0 . Since y 2 = 0 , it follows from Lemma 1 that there exist two elements a , b R such that a + b = 1 , r y = 0 , and b y = y b = 0 . Then, y = y ( a + b ) = y a + y b = 0 . 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 r , s R such that r s = 0 ; see ([8], p. 234, Corollary). The uniserial ring R is local; therefore, the invertible elements of R form the Jacobson radical J ( R ) 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 a , b R such that a + b = 1 , r a = 0 , and b s = 0 . We have that either a R b R or b R a R ; in addition, a R + b R = R = R a + R b . Therefore, at least one of the elements a , b 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 r a = 0 and b s = 0 . □
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 x y y x 0 for some x , y R . We note that A = { c C : x c C } is an ideal of the ring C. The set d C | d A = 0 is not empty, since we can take d = 0 . We take any element d C with d A = 0 . If x d 0 , then x d z C \ { 0 } for some z C . Hence d z A , and therefore, d ( d z ) = 0 and ( d z ) 2 = 0 . Thus, d z = 0 and x d z = 0 ; this is a contradiction. Therefore, x d = 0 , and thus, d A . Therefore, d 2 = 0 and d = 0 . This implies that Ann C ( A ) = 0 . For any a A , we have that x a = a x C . Thus,
( x y y x ) a = x ( y a ) y ( x a ) = x a y x a y = 0
and ( x y y x ) A = 0 . However, c 1 ( x y y x ) = c 2 for some nonzero elements c 1 , c 2 C , so c 2 A = 0 and, hence, Ann C ( A ) 0 ; 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. □

Funding

The work of Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013P.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Conflicts of Interest

The author declares no conflict of interest.

References

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Tuganbaev, A. On Rings of Weak Global Dimension at Most One. Mathematics 2021, 9, 2643. https://doi.org/10.3390/math9212643

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Tuganbaev A. On Rings of Weak Global Dimension at Most One. Mathematics. 2021; 9(21):2643. https://doi.org/10.3390/math9212643

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Tuganbaev, Askar. 2021. "On Rings of Weak Global Dimension at Most One" Mathematics 9, no. 21: 2643. https://doi.org/10.3390/math9212643

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