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 × 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 Benz [3] describes classical geometries of Möbius, Laguerre and Minkowski, using the notion of the projective line over a ring. Veldkamp in [19] 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, Herzer [11] defines a point of the projective line as a cyclic submodule generated by an admissible pair. Hence points of P(R) are elements of an 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 existence of points of P(R) properly contained in another point. Blunck and Havlicek remark that avoiding this bizarre situation is equivalent to the assumption that a ring is Dedekind-finite, see [6,Proposition 2.2].
Havlicek and Saniga [15] propose to consider another type of free cyclic submodules, i.e. represented by pairs not contained in any cyclic submodule generated by a 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 3 × 3 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 from the definition.
Using the classification of finite rings from [9,10], 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, i.e. R(a, b) is non-torsion. We assume that R satisfies the invariant basis property (IBP) [8]. For such rings the basis of a cyclic submodule R(a, b) ⊂ R 2 is always of cardinality 1 and any invertible matrix with entries in R has square size, i.e. it belongs to the general linear group GL n (R) for some natural number n.
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. This leads to a definition of 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 an earlier definition of the projective line over commutative ring used by Benz [3], the points of the projective line are cyclic submodules generated by unimodular pairs.
From now on, whenever we will write 'unimodularity', we always mean 'right unimodularity'. We also call the cyclic submodule R(a, b) unimodular, if (a, b) is. 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, other 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.
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. 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) generators.
We substitute 'admissible' by 'unimodular' in [6, Proposition 2.1] and we get: If (a, b) is unimodular, then there exist a , b ∈ R with aa +bb = 1, which 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 any cyclic submodule R(a, b) that is generated by a unimodular (resp. admissible) pair does not have nonunimodular (resp. non-admissible) generators.
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 nonzero 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. unimodular and (a, b) = r(x, y). Then the following are equivalent: Remark 3. Let R be a ring and let (a, b) ∈ R 2 . If there exists (x, y) ∈ R 2 and a left zero divisor r ∈ R such that (a, b) = r(x, y), then R(a, b) is not a free cyclic submodule.
Proof. Suppose that the above assumptions are satisfied. Hence there exists nonzero α ∈ R such that αr = 0, which yields: Condition (F ) implies that the same free cyclic submodule can be generated by two pairs precisely when they are left-proportional by an invertible element of R. If a pair (a, b) ∈ R 2 generates a free cyclic submodule, then all cyclic submodules that contain (a, b) are free.
Proof. Assume that there exist (x, y) ∈ R 2 and r ∈ R such that (a, b) = r(x, y). According to Remark 3. this r is an invertible element of R, hence R(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 Havlicek and Saniga in [15].

Definition 4 [12, Definition 9]. A pair (a, b) ∈ R 2 that is not contained in any cyclic submodule generated by a unimodular pair is called an outlier.
In the last section, will be needed one more concept. Recall that a monomorphism of modules f :

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 non-principal ideals [12].
Corollary 3 [12,Theorem 13]. Let (a, b) ∈ R 2 be non-unimodular. Then (a, b) is an outlier, if one of the following conditions is satisfied: 1. There does not exist a principal proper right ideal αR such that a, b ∈ αR.
2. aR+bR αR for all principal proper right ideals αR such that a, b ∈ αR. If there exists a principal proper right ideal αR containing a and b, then  R(a, b) is a torsion cyclic submodule.

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 if (a, b) = r(x, y) for some (x, y) ∈ R 2 , which follows from Remark 3. Hence (a, b) is an outlier (see Corollary 2). Example 3 [14,15]. Consider the ring T of ternions over the commutative field F , i.e. the ring of upper triangular 2 × 2 matrices with entries from F . The free cyclic submodules of T 2 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 3 × 3 matrices with entries from an arbitrary commutative field F . We consider now all possibilities: . From 1. we get f = f = 0. In the same manner (considering other cases and multiplying by invertible matrices) we get all orbits of pairs generating free cyclic submodules of T 2 3 . It is easy to check that they are distinct, i.e. there is no invertible matrix that converts one orbit to another. Multiplying the representatives of the orbits with invertible elements of T 3 (from the left) immediately yields that all free cyclic submodules generated by pairs from point 3. are in the same orbit: