Free cyclic submodules in the context of the projective line

We discuss the free cyclic submodules over an associative ring $R$ with unity. Special attention is paid to those, which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular $3$x$3$ matrices over a field $F$ under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.


Introduction
In [3] W. Benz describes classical geometries of Möbius, Laguerre and Minkowski, using the notion of the projective line over a ring. F.D. Veldkamp in [18] points out that the assumption of a ring to be of stable rank 2, allows to generalize many properties from classical projective geometry over a field. They both define the projective line by unimodular pairs. To admit a wider class of rings, A. Herzer [11] defines a point of the projective line as a cyclic submodule generated by admissible pair. Hence points of P(R) are elements of the orbit under the action of the GL 2 (R). In the present paper we adopt this convention as well. This approach without any assumptions leads to the existance of points of P(R) properly contained in another point. A. Blunck and H. Havlicek remark that avoiding this bizarre situation is equivalent to the assumption that a ring is Dedekind-finite, see [6,Proposition 2.2]. H. Havlicek and M. Saniga [15] propose to consider another type of free cyclic submodules, i.e. represented by pairs not contained in any cyclic submodule generated by an unimodular pair (so-called outliers). In this note, we show that the class of non-unimodular free cyclic submodules can be wide. We find four orbits of such submodules in the ring T 3 of lower triangular 3x3 matrices over a field F under the action of GL 2 (T 3 ). On the other hand there are classes of rings without outliers (e.g. semisimple rings, Proposition 5.) and rings such that outliers generate only torsion submodules (e.g. finite commutative rings, Theorem 5.). This answers the question posed in [12] about outliers in finite rings. We remark also that there are infinite rings with non-unimodular free cyclic submodules properly contained in unimodular ones (Proposition 4). Furthermore, we show that if R is a finite ring and non-unimodular R(a, b) ⊂ R 2 is free, then (a, b) is an outlier. In that case, there is no need to check the condition resulting from the definition. Using the classification of finite rings, we find all rings up to order p 4 , p − prime, with outliers generating free cyclic submodules. The problem to completely characterize rings with outliers, especially generating free cyclic submodules, is still open.

Preliminaries
Throughout this paper we shall only consider associative rings with 1 (1 = 0). The group of invertible elements of the ring R will be denoted by R * . If R is a ring, the expression R 2 will mean a left free module over R. If (a, b) ∈ R 2 , then the set: R(a, b) = {(αa, αb) : α ∈ R} is a left cyclic submodule of R 2 . It is called free if the equation (ra, rb) = (0, 0) implies that r = 0. We assume that R satisfies invariant basis property (IBP) [8]. For such rings the basis of cyclic submodule R(a, b) ⊂ R 2 is always of cardinality 1 and any invertible matrix is in the general linear group GL n (R) of invertible matrices with entries in R. The general linear group GL 2 (R) acts in natural way (from the right) on the free left R-module R 2 .
Definition 1 [6] The projective line over R is the orbit of the free cyclic submodule R(1, 0) under the action of GL 2 (R).
In other words, the points of P(R) are those free cyclic submodules of R 2 which possess a free cyclic complement. It provides to introduce admissibility.
i.e. R(a, b) is a free cyclic submodule which has a free cyclic complement. If R is commutative, then the condition mentioned above is equivalent to Therefore P(R) = {R(a, b) ⊂ R 2 , (a, b) admissible}. As we mentioned before, in earlier definition of the projective line over commutative ring used by W. Benz [3], the points of the projective line are cyclic submodules generated by unimodular pairs. Definition 3 A pair (a, b) ∈ R 2 is right unimodular, if there exist elements x, y ∈ R such that ax + by = 1.
From now on, whenever we will write 'unimodularity', we always mean 'right unimodularity'.
Remark 1 Obviously, the admissibility implies the unimodularity and if (a, b) ∈ R 2 is unimodular, then R(a, b) is a free cyclic submodule of R 2 .
In contrast to the cyclic submodules generated by admissible pairs, others free cyclic submodules do not have a free cyclic complement in R 2 .
The following simple remark describes unimodularity in terms of (right) ideals.
Remark 2 Let R be a ring and a, b ∈ R. The following statements are equivalent: 1. aR + bR = R.
2. There exist elements x, y ∈ R such that ax + by = 1.
3. There is no proper right ideal I such that a, b ∈ I.
2. ⇒ 3. Suppose that there exist x, y ∈ R such that ax + by = 1 and let I be a right ideal such that a, b ∈ I. Of course, ax ∈ I for all x ∈ R and by ∈ I for all y ∈ R. Consequently (ax + by) ∈ I for all x, y ∈ R. Thus 1 ∈ I, and therefore I = R.
3. ⇒ 2. Assume that ax + by = 1 for all x, y ∈ R, then {ax + by; x, y ∈ R} = aR + bR = R. aR, bR are ideals, thus aR + bR is an ideal too, and it contains a, b. So, aR + bR is a proper right ideal which contradicts 3.

⊓ ⊔
In general, cyclic submodules generated by unimodular (resp. admissible) pairs can be also generated by non-unimodular (resp. non-admissible) ones. In special cases cyclic submodule generated by unimodular (resp. admissible) pair cannot have non-unimodular (resp. non-admissible) representation.
We substitute 'admissible' by 'unimodular' in [6, Proposition 2.1 (2)] and we get: " ⇒ " If r is right invertible, then rs = 1 for some s ∈ R. Hence (a, b) is unimodular: implies that r has a right inverse:

⊓ ⊔
Rings with the property ab = 1 ⇒ ba = 1 are called Dedekind-finite. On account of the above proposition and of the Proposition 2.1 (2) [6] we obtain: Corollary 1 If R is Dedekind-finite, then the cyclic submodule R(a, b) generated by an unimodular (resp. admissible) pair do not have non-unimodular (resp. nonadmissible) representation.
As we know, each admissible pair (a, b) ∈ R 2 is unimodular. What about the converse implication? There are examples of rings where unimodularity does not imply admissibility [7, Remark 5.1]. However, it is also known that if R is a ring of stable rank 2 (for example, local rings and matrix rings over fields), then admissibility and unimodularity are equivalent and R is Dedekind-finite [6, Remark 2.4]. So finite or commutative rings satisfy this property as well. In case of such rings, the projective line can be described by using unimodularity or admissibility interchangeably.
Let us introduce the following temporary notation: (F ) Any element of the ring R is either invertible or a zero divisor. It is known that any finite ring satisfies (F ). Additionally, if R satisfies (F ), then the ring M n (R) (n 1) fulfills this condition as well.
Corollary 2 Let R satisfy (F ), (x, y) ∈ R 2 be unimodular and (a, b) = r(x, y). Then the following are equivalent: is not a free cyclic submodule.

⊓ ⊔
Condition (F ) implies that the same free cyclic submodule can be represented by two pairs exactly if they are left-proportional by an invertible element of R.
Proposition 2 Let R satisfy (F ). A pair (a, b) ∈ R 2 generating a free cyclic submodule is contained solely in free cyclic submodules.
Proof. Assume that there exist (x, y) ∈ R 2 and r ∈ R such that (a, b) = r(x, y).
The next class of cyclic submodules, which can be considered in the context of the projective line, is the one proposed by H.

Definition 4 [12, Definition 9]
A pair which is not contained in any cyclic submodule generated by an unimodular pair is called an outlier.
In the last section, will be needed one more concept. Recall that the monomor- In other words, sequence of left modules 0 −→ M ′ −→ M is split.

Free cyclic submodules generated by non-unimodular pairs
For some rings there are free cyclic submodules which are generated only by non-unimodular pairs. We establish some connections between outliers and nonprincipal ideals, [12].
Proposition 3 Let (a, b) ∈ R 2 be non-unimodular. If aR+bR is a non-principal right ideal, then (a, b) is an outlier.
2. aR + bR αR for all principal proper right ideals αR such that a, b ∈ αR.
1. If there exists a principal proper right ideal αR containing a and b, then R(a, b) is a torsion cyclic submodule. 2. If R(a, b) ⊂ R 2 is non-unimodular and free, then (a, b) is an outlier.
Proof. 1. Suppose that the above assumptions are satisfied. Then there exist c, d ∈ R such that a = αc, b = αd and nonzero r ∈ R with rα = 0. We thus get r(a, b) = r(αc, αd) = (0, 0), which is our claim. 2. If R(a, b) is a free cyclic submodule, then r ∈ R is invertible for all (x, y) ∈ R 2 such that (a, b) = r(x, y), which follows from Remark 3. Hence (a, b) is an outlier. Example 3 [14] Consider the ring T of ternions, which is isomorphic to the ring of upper triangular 2x2 matrices with entries from an arbitrary commutative field F . The free cyclic submodules fall into two distinct orbits under the action of the GL 2 (T ): .
The first orbit makes up the projective line P(T ), the second one is the orbit of free cyclic submodules generated by outliers.
Let R = T 3 be the ring of lower triangular 3x3 matrices with entries from an arbitrary commutative field F . .
We obtain from 2. that a = 0 or a ′ = 0, and then we assume a = 0. Multiplying by the invertible matrix We consider now all possibilities: which gives a pair In the same manner (considering other cases and multiplying by invertible matrices) we get all orbits of pairs generating free cyclic submodules of T 3 . It is easy to check that they are distinct, i.e. there is no any invertible matrix that converts one orbit to another. Multiplication (from the left) representatives of orbits by the invertible elements of T 3 , follows immediately that free cyclic submodules generated by pairs from point 3. are in the same orbit:

⊓ ⊔
Corollary 4 Two pairs (x, y), (w, z) ∈ T 3 2 generating free cyclic submodules are in the same GL 2 (T 3 )-orbit if, and only if, the right ideals generated by x, y ∈ T 3 and by w, z ∈ T 3 coincide.
Proof. Let I (x,y) denote the right ideal of T 3 which is generated by x and y. " ⇒ " If pairs (x, y), (w, z) ∈ T 3 2 are in the same GL 2 (T 3 )-orbit, then there exists a matrix A ∈ GL 2 (T 3 ) such that (x, y)A = (w, z). This gives I (w,z) ⊆ I (x,y) . Next we multiply last equation by A −1 , which yields (x, y) = (w, z)A −1 , and, in consequence I (x,y) ⊆ I (w,z) . The result is I (x,y) = I (w,z) . " ⇐ " This is straightforward from Theorem 2.
⊓ ⊔ In case of rings without (F ) there are also non-unimodular free cyclic submodules, that are not generated by outliers.

Proposition 4
If R is a (commutative) PID, then non-unimodular free cyclic submodules are generated by non-outliers.
Proof. We use the following equivalent characterization of a proper (commutative) PID: prime ideals are maximal if they are nonzero; -prime ideals are principal; -gcd(a, b) = 1 ⇒ gen(a, b) = 1 for any a, b ∈ R; -R is Bezout.

Rings without non-unimodular free cyclic submodules
There are some rings R such that free cyclic submodules R(a, b) are generated only by admissible pairs (a, b) ∈ R 2 . They all makes up the projective line P(R), for instance, fields or finite local rings [12,Theorem 20. 1]. In case of these rings all free cyclic submodules can be written as a projective line: Corollary 5 If R is a commutative finite ring, then all free cyclic submodules make up the projective line P(R).
Example 4 Let us consider the finite noncommutative ring R of characteristic p 2 , p − prime. The additive group of R is equal to R + = Z p 2 ⊕ Z p ⊕ Z p with a basis {1, t, y}. The multiplication in the ring R is uniquely determined by the relations t 2 = 0, y 2 = y, ty = 0, yt = t, [9].
We have (1 − t − y)(r + st + hy) + t(r ′ + s ′ t + h ′ y) = r + (−r + r ′ )t − ry for some 0 r, r ′ p 2 − 1, 0 s, h, s ′ , h ′ p − 1, hence the pair (1 − t − y, t) is nonunimodular. It is easily seen that R(1 − t − y, t) is free. On account of Theorem 1. 2. (1 − t − y, t) is an outlier. R is an example of a ring non-embeddable into the ring of matrices over GF (p 2 ), [17]. Now we are able to describe all finite rings up to order p 4 , p − prime, with outliers generating free cyclic submodules. Since any finite ring with identity is isomorphic to direct sum of rings with identity of prime power order (see [16]) and according to Theorems 4. and 5., we may restrict ourselves to the study noncommutative indecomposable rings up to order p 4 , p−prime. By direct calculation, taking into account classification theorems ( [10], [9]) we find that there are exactly four such rings for any p: of order p 3 : the ring of ternions over GF (p); -of order p 4 : • with characteristic p: • with characteristic p 2 -the ring from the Example 4.