On Rings of Weak Global Dimension at Most One

: 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 xy = 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.


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 → XY 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 R 2 (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 ∈ rRr 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 rx = 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/2Z and Q 8 is the quaternion group of order 8, then the group algebra FQ 8 is a finite noncommutative centrally essential ring; see [5].
Let F be the field Z/3Z, 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 an arithmetical semiprime ring

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: The ring R is not semiprime, since the following set is a nonzero nilpotent ideal of R: . 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.

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 rs = 0, then there exist two elements a, b ∈ R such that a + b = 1, ra = 0, and bs = 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 rx = y. Since r 2 = 0, we have that y 2 = (rx) 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, ry = 0, and by = yb = 0. Then, y = y(a + b) = ya + yb = 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 rs = 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, ra = 0, and bs = 0. We have that either aR ⊆ bR or bR ⊆ aR; in addition, aR + bR = R = Ra + Rb. 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 ra = 0 and bs = 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 xy − yx = 0 for some x, y ∈ R. We note that A = {c ∈ C : xc ∈ C} is an ideal of the ring C.
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.