A short note on strongly flat covers of acts over monoids
aa r X i v : . [ m a t h . G R ] O c t A short note on strongly flat covers of acts over monoids
Alex Bailey and James RenshawSeptember 2013
Abstract
Recently two different concepts of covers of acts over monoids have been studied. Thatbased on coessential epimorphisms and that based on Enochs’ definition of a flat coverof a module over a ring. Two recent papers have suggested that in the former case,strongly flat covers are not unique. We show that these examples are in fact falseand so the question of uniqueness appears to still remain open. In the latter case, were-present an example due to Kruml that demonstrates that, unlike the case for flatcovers of modules, strongly flat covers of S − acts do not always exist. Key Words
Semigroups, monoids, acts, strongly flat, covers.
Let S be a monoid. By a right S − act we mean a non-empty set X together with an action X × S → X given by ( x, s ) xs such that for all x ∈ X, s, t ∈ S, x x and x ( st ) = ( xs ) t .We refer the reader to [7] for basic results and terminology in semigroups and monoids andto [1] and [9] for all undefined terms concerning acts over monoids.Enochs’ conjecture, that all modules over a unitary ring have a flat cover, was finally provenin 2001. In 2008, Mahmoudi and Renshaw [13] initiated a study of flat covers of acts overmonoids. Their definition of cover proved to be different to that given by Enochs and in2012, Bailey and Renshaw [2] initiated a study of Enochs’ definition of cover. We give hereboth definitions but note that we use a slightly different terminology to that used in [13]. Definition 1.1
Let S be a monoid, A an S − act and let X be a class of S − acts closed underisomorphisms.1. We shall say that an S − act C together with an S − epimorphism f : C → A is a coessential-cover of A if there is no proper subact B of C such that f | B is onto. If inaddition C ∈ X then we shall call it an X − coessential-cover .2. By an X - precover of A we mean an S − map g : P → A for some P ∈ X such thatfor every S − map g ′ : P ′ → A , for P ′ ∈ X , there exists an S − map f : P ′ → P with University of Southampton, Southampton, SO17 1BJ, EnglandEmail: [email protected]@maths.soton.ac.uk ′ = gf . P AP ′ f g ′ g If in addition the
X − precover satisfies the condition that each S − map f : P → P with gf = g is an isomorphism, then we shall call it an X − cover .It was shown in [3, Lemma 2.1] that if X is the class of projective (or free) S − acts thenthese two definitions coincide, whereas if S is the infinite monogenic monoid and X = SF is the class of strongly flat S − acts then it easily follows from [13, Corollary 3.3] that the1-element S − act does not have an SF− coessential-cover but does have an
SF − cover by [2,Corollary 5.6].It is also easy to show that
X − covers, when they exist, are unique up to isomorphism,whereas this is not true, in general, for
X − coessential-covers. However the question as tothe uniqueness of
SF− coessential-covers has remained open, as apparently has the questionof whether all S − acts have an SF − cover.
Recently Qiao and Wei have published an example of a monoid that they claim demonstratesthat some S − acts do not have unique SF − coessential-covers [12, Example 2.5]. Howevertheir result is false.Let S = h x , x , . . . | k ≥ , x x k = x k x = x , x kk = x k +1 k ; i, j ≥ , x i x j = x j i ∪ { } and define a right S − congruence ρ on S by ( s, t ) ∈ ρ if and only if either ( s, t ) ∈ h x i or s, t ∈ h x , x , . . . i ∪ { } . For notational convenience, let us denote 1 by x i for any i ≥ R i = h x i i ∪ { } and define a right congruence σ i on S by ( s, t ) ∈ σ i if and onlyif there exists p, q ∈ R i with ps = qt . It is clear that if i = j then σ i = σ j . Qiao andWei claim that S/σ i and S/σ j are distinct SF− coessential-covers for
S/ρ . However by[13, Lemma 2.4] we see that
S/σ i ∼ = S/σ j if and only if there exists u ∈ S such that σ i = { ( s, t ) ∈ S × S | ( us, ut ) ∈ σ j } . We show now that this is indeed the case.First let us suppose without loss of generality that j > i ≥
0. Suppose also that ( s, t ) ∈ σ i so that there exists p, q ∈ R i with ps = qt . Assume without loss of generality that s = t .We consider two cases:1. Suppose that i = 0. Then σ = S × S = { ( s, t ) ∈ S × S | ( x s, x t ) ∈ σ j } .2. Suppose that i > s = 1. Then t ∈ h x i i and so x jj x i s = x jj x i t .(b) If s ∈ h x k i , k ≥ t ∈ R k (notice that if t = 1 then k = i ) and so x kj x i s = x kj x i t .In both cases we deduce that ( s, t ) ∈ { ( s, t ) ∈ S × S | ( x i s, x i t ) ∈ σ j } .Conversely, if ( x i s, x i t ) ∈ σ j then there exists p ′ , q ′ ∈ R j with p ′ x i s = q ′ x i t . But p ′ x i , q ′ x i ∈ h x i i and so ( s, t ) ∈ σ i . 2onsequently S/σ i ∼ = S/σ j and the SF − coessential-covers are not distinct.Notice in fact that by [5, Proposition 2.8]
S/ρ does indeed have a unique
SF− coessential-cover since every element of the form x jj is a right zero in [1] ρ .There is a similar mistake to be found in [11, Example 3.4] where the same claim is made.Indeed the monoid in this example is finite whereas Ershad and Khosravi in [5] give anextensive list of monoids, which include the finite ones, where SF − coessential-covers areunique, when they exist. A much simpler example with the same property would be themonoid S = R where R is any right zero semigroup. Given any z ∈ R , define R z = { , z } and ( s, t ) ∈ σ z if and only if there exists p, q ∈ R z such that ps = qt . Then for every z , z ∈ R , σ z = σ z and S/σ z and S/σ z are both SF − coessential-covers of the 1 − element S − act. However, S/σ z and S/σ z are both isomorphic. Consequently the question of when SF− coessential-covers are unique would appear still to be an open one.
S F -covers
Enochs, Bican and El Bashir finally proved that all modules over a ring have flat covers in2001. Similar results have subsequently been proved in a number of other categories andthe obvious analogue for acts over monoids would be the existence of
SF− covers. Howeverit has recently been brought to our attention that Kruml [10] has provided an example toshow that
SF− covers do not always exist. The result below is essentially Kruml’s, ourcontribution being to translate the proof from the language of varieties to the language of S − acts. We also pose a new question about the existence of SF− covers.
Proposition 3.1 (Cf. [10, Proposition 3.1])
Let T = h a , a , a · · · | a i a j = a j +1 a i for all i ≤ j i and let S = T , then the one element S -act Θ S does not have an SF -precover. Proof.
We first note that S is left cancellative. In fact, every word w ∈ T has a uniquenormal form w = a α (1) · · · a α ( n ) where α ( i ) ≤ α ( i + 1) for all 1 ≤ i ≤ n −
1, and given any a α ( n +1) , a β ( n +1) , it is easy to see that wa α ( n +1) = wa β ( n +1) implies α ( n + 1) = β ( n + 1).Hence every S − endomorphism h : S → S is injective, as h ( s ) = h ( t ) implies h (1) s = h (1) t .Assume Θ S does have an SF -precover, then by [2, Lemma 4.7], SF contains a weaklyterminal object, say T . By [14, Theorem 5.3], let ( T, α i ) i ∈ I be the directed colimit offinitely generated free S -acts ( T i , φ i,j ) i ∈ I . Let X be any totally ordered set with | X | > max {| I | , ℵ , | S |} and let Fin( X ) denote the set of all finite subsets of X . We now define adirect system indexed over Fin( X ) partially ordered by inclusion, where every object S Y isisomorphic to S and a map from an n − Y into an n element subset Y ∪ { z } is defined to be the endomorphism λ a i : S → S , s a i s , where i = |{ y ∈ Y | y < z }| . Itfollows from the presentation of S that this is indeed a direct system. Let ( F, β Y ) Y ∈ Fin( X ) be the directed colimit of this direct system, which by [14, Proposition 5.2], is a stronglyflat act. Therefore, there exists an S -map t : F → T . Now for each singleton { x } ∈ Fin( X ), tβ { x } (1) ∈ T and so there exists some i ∈ I and x i ∈ T i such that α i ( x i ) = tβ { x } (1). Definethe S -map θ i : S { x } → T i , s x i s and then tβ { x } = α i θ i . So by the axiom of choice wecan define a function h : X → Z, x ( i, x i ) where Z := { ( i, x ) ∈ { i } × T i | i ∈ I } and | Z | ≤ max {| I | , ℵ , | S |} . Since | X | > | Z | , h cannot be an injective function and so there exist x = y ∈ X with h ( x ) = h ( y ). Since θ i is determined entirely by the image of 1, we have that tβ { x } = α i θ i = tβ { y } . Without loss of generality, assume x < y in X , then β { x,y } λ a = β { x } β { x,y } λ a = β { y } . Similarly there also exists j ∈ I , θ j ∈ Hom( S { x,y } , T j ) such that tβ { x,y } = α j θ j . Therefore we have α i θ i = tβ { x } = tβ { x,y } λ a = α j θ j λ a ⇒ α i ( θ i (1)) = α j ( θ j λ a (1))and so by [2, Theorem 2.2],there exists some k ≥ i, j such that φ i,k ( θ i (1)) = φ j,k ( θ j λ a (1))which implies φ i,k θ i = φ j,k θ j λ a . Similarly α i θ i = tβ { y } = tβ { x,y } λ a = α j θ j λ a = α k φ jk θ j λ a ⇒ α i ( θ i (1)) = α k (cid:16) φ jk θ j λ a (1) (cid:17) which again, implies there exists some m ≥ i, k such that φ i,m θ i = φ k,m φ j,k θ j λ a = φ j,m θ j λ a . Therefore φ j,m θ j λ a = φ k,m φ j,k θ j λ a = φ k,m φ i,k θ i = φ i,m θ i = φ j,m θ j λ a . Since both T j and T m are finitely generated free S -acts, and S { x,y } is a cyclic S -act, it isclear that φ j,m θ j is an endomorphism of S and so a monomorphism. Therefore λ a = λ a which implies a = a which is a contradiction.It is still an open question to find necessary and sufficient conditions for the existence of SF− covers for the category of acts over monoids.Recall that a ring/monoid is called right perfect if every right module/act over it has aprojective cover. Bass proved in 1960 that a ring is right perfect if and only if it satisfies M L , the descending chain condition on principal left ideals [4]. It was shown in [8] and [6]that the case for monoids is different. A monoid is right perfect if and only if it satisfies M L and Condition ( A ), every right S − act has the ascending chain condition on cyclic subacts.It was shown in [2, Proposition 5.7] that a monoid S satisfying Condition ( A ) is a sufficientcondition for every S − act to have an SF − cover. The converse however is not true, theinfinite monogenic monoid being a counterexample. It was also shown in [2, Corollary 5.6]that S being right cancellative is sufficient for every S − act to have an SF -cover, and as wecan see from the next Lemma, right cancellativity and M L implies Condition ( A ). Lemma 3.2
A right cancellative monoid with M L is a group. Proof.
Given any s ∈ S , consider the chain Ss ⊇ Ss ⊇ Ss ⊇ . . . , by M L there existssome n ∈ N such that Ss n = Ss n +1 and by right cancellativity this implies S = Ss .Since the only known counterexample to the existence of SF− covers, as given above, doesnot have M L , it seem natural to pose the following question Question 3.3
Is it true that a monoid S is right perfect if and only if it satisfies M L andevery right S − act has an SF -cover? This clearly generalises the situation for rings where modules always have flat covers and aring is perfect if and only if it has M L . Clearly one way is obvious as a perfect monoid hasCondition ( A ) and so every S − act has an SF − cover. The converse however is not clear. Wewould like to know if there exists a monoid S satisfying M L but not Condition ( A ) for whichevery S − act has an SF− cover. Another class of monoids known to have
SF − covers arethose monoids having weak finite geometric type (see [2, Proposition 5.4]). This would seemto be a good place to look for a counterexample, although the main example of a monoidhaving weak finite geometric type, that is not right cancellative, is the Bicyclic monoid whichdoes not have M L . 4 eferences [1] Ahsan, J., Zhongkui, L. A Homological Approach to the Theory of Monoids , SciencePress, Beijing, (2008).[2] Bailey, A., Renshaw, J., Covers of acts over monoids and pure epimorphisms,
Proc.Edinburgh Math. Soc. , to appear, arXiv:1206.3095.[3] Bailey, A., Renshaw, J. Covers of acts II, to appear in
Semigroup Forum .[4] Bass, H., Finitistic dimension and a homological generalization of semi-primary rings,
Trans. Amer. Math. Soc.
95 (1960) 466–488.[5] Ershad, M., Khosravi, R., On the uniqueness of strongly flat covers of cyclic acts,
TurkJ Math
35 (2011) 437–442.[6] Fountain, J., Perfect semigroups,
Proc. Edinburgh Math. Soc
20 (1976) 87–93.[7] Howie, J.M.
Fundamentals of Semigroup Theory , London Mathematical Society Mono-graphs, (OUP, 1995).[8] Isbell, John R., Perfect monoids,
Semigroup Forum
Monoids, Acts and Categories , De Gruyter Ex-positions in Mathematics (29) , (Walter de Gruyter, Berlin, New York, 2000).[10] Kruml, D., On flat covers in varieties,
Comment. Math. Univ. Carolin.
49 (2008), 19–24.[11] Qiao, H., Wang, L., On flatness covers of cyclic acts over monoids,
Glasg. Math. J. _ releasepaper/downPaper/201212-280 (2012).[13] Mahmoudi, M., Renshaw, J., On covers of cyclic acts over monoids, Semigroup Forum (2008) 77, 325–338.[14] Stenstr¨om, B., Flatness and localization over monoids,