The core ingredient of the definition of a Frobenius algebra finitely generated as a module over its commutative ground ring , as it appears in the literature, is the requirement that there exist a non degenerate associative –bilinear form on ; this requirement mimics accurately the definition of a Frobenius algebra over a field. However, that definition also includes the somewhat technical requirement that be projective as a –module. At first, since all modules over a field are indeed free and hence projective, the extra condition appears inocuous. Furthermore, the extra condition apparently served the purpose of affording the result, mentioned without a detailed proof in [7, p. 434], that every Frobenius algebra over a Frobenius commutative ring is a Frobenius ring. Unfortunately, not every finite Frobenius ring is projective over its characteristic subring.
In this paper we remove the projectivity technical requirement from the usual definition of a Frobenius algebra over a commutative ring and propose that said expression be used instead for a properly modified notion. We show that thus extending the definition affords us a very natural result, namely, that a finite ring of characteristic is Frobenius if and only if it is a (not necessarily projective) Frobenius algebra over its characteristic ring . We, therefore, adopt the expression non projective Frobenius algebra over a ring when all else is satisfied but possibly not the projectivity requirement. Notice that, as is a common practice in mathematics, our expression “non projective”, is a less pedantic option to express “not necessarily projective”.
An additional benefit from our approach is that we may adapt, without using the projectivity of over , the arguments in the usual proof that Frobenius algebras over a field are Frobenius as rings to the case of non projective Frobenius algebras over a ring (Theorem 16). This result is interesting in its own right and has potential to open many doors in the realm of applications. Note that one may also derive Theorem 16 from the recent analysis of the Frobenius property for Artin algebras in . However, while more general, this approach is significantly more technical and we felt that there is value in sharing our proof here in hopes that it would be more accesible for a larger group of interested readers.
One of the motivations behind this extension of the expression non projective Frobenius algebra over a ring arises because of applications of this notion in the algebraic theory of error-correcting codes. Finite Frobenius rings are extensively used as alphabets for ring- linear block codes; Frobenius algebras over a finite field are an important example of Frobenius rings. In contrast, a well known characterization of Frobenius rings  may be rephrased by asserting that a finite ring is Frobenius if and only if there exists an non degenerate associative bilinear form over its characteristic subring.
The fundamental problem in the theory of linear codes is the analysis and classification of linear codes over an alphabet, usually a finite field
. Since there is a tremendous number of subspaces for the vector spacethe task at hand is quite monumental. A common frequently successful practice has been to filter the codes considered in the following manner: add an -algebra structure on the vector space (let us refer to , equipped with the additional structure, as to remind us of the fact that it is now an algebra of dimension ) and consider, say, only those subspaces which are (left, right, two-sided) ideals of . We refer to the additional algebraic structure on as the ambient (or the ambient space) of the codes targetted and to the criterion applied to choose the codes we wish to consider as the filter being applied.
As the interest in studying codes in ambients endowed with additonal algebraic structure expanded from considering field alphabets to ring alphabets, Frobenius rings have tended to become central to the conversation. The reason for this preference goes back to papers that show that the fundamental structural properties of codes over fields extend to codes that have Frobenius rings as their alphabets. In fact, there is plenty of evidence to show that Frobenius rings are precisely the extent to which this type of considerations can be extended if one wishes to acquire the desired structural properties. For the properties to hold, the alphabet must be Frobenius.
In this paper, we investigate an alternative role that Frobenius rings appear to play very naturally in this context. It seems to be the case that coding-theoretic structural properties are attained, regardless of the structure of the alphabet, when the ambient is a non projective Frobenius algebra over a commutative ring possibly other than the alphabet. For example, it may be that the ambient is a non projective Frobenius algebra over its characteristic subring.
We illustrate our ideas by taking advantage of the fact that every finite Frobenius ring of characteristic can be seen as a non projective Frobenius
–algebra. Our examples include constructing a family of non projective Frobenius algebras based on factor algebras of skew polynomial algebras and using them as the ambient algebra for a wide class of analogues to skew cyclic block codes.
This note is organized as follows. Section 2 provides, with the generality needed here, the statements and proofs of preliminary results on balanced bilinear forms over bimodules; we only include proofs that we deemed illustrative. Section 8 provides our proposed definition of a non projective Frobenius algebra over a commutative ring and various equivalent characterizations; as mentioned above, projectivity of the algebra over the base ring is not required. One of these equivalent conditions is the existence of a Frobenius functional, which will be seen to generalize the generating character and play a similar role to it. In Section 4, we extend results from  on annihilators associated to a non degenerate bilinear form from a finite Frobenius ring to a non projective Frobenius algebra. Section 5 contains our observation that, given an algebra over a Frobenius commutative ring such that is finitely generated as a –module, then is a non projective Frobenius algebra over if and only if is a Frobenius ring. This, in particular, applies to finite rings, viewed as algebras over their characteristic subrings. We also include a method, based on skew polynomials, to construct new Frobenius algebras from a given one. From the point of view of codes, this gives a way to construct new finite Frobenius rings from old ones; this is further discussed in Section 6. That section also contains a discussion of how the general results on bilinear forms defined on modules over non projective Frobenius algebras, developed in the previous sections, may be applied to get the main results of . We close that section with a discussion of for which bilinear forms the version of McWilliams identities stated in  holds.
2. Preliminaries on bilinear forms and annhililators
All rings we consider will be unital and possibly non commutative. Let rings and an –bimodule. We will simply say that is a bimodule. We also use the notation to declare that is a left –module, and for right –modules. Systematic introductions the theory of modules over non commutative rings are , or . Everything in this sectiond follows from [1, §30]. For convenience, however, we state the results here in the form they are needed, and provide their proofs in this more limited scope to keep the paper self-contained. Given a second bimodule , the abelian group of all homomorphisms of right –modules is endowed with the structure of a bimodule by the rule
A straightforward argument shows that the formula
provides a bijective correspondence between –bilinear maps
and homomorphisms of left –modules
Moreover, is associative, in the sense that for all , if and only if the corresponding is a homomorphism of right –modules.
Analogously, the additive group of all homomorphisms of left –modules is a –bimodule via the rule
gives a bijective correspondence between –bilinear forms
and homomorphisms of right –modules
Again, the condition of being the bilinear form associative is equivalent to require that is a homomorphism of left –modules.
Let be an –bilinear form, and , the corresponding homomorphisms of –modules defined as before. We say that is right (resp. left) non degenerate if (resp. ) is injective. When is left and right non degenerate, we just say that the bilinear form is non degenerate.
The length of a right –module will be denoted by , for a left –module , by . From now on, unless otherwise stated, we will assume that all –modules are of finite length. We also assume in the rest of this section that is a Quasi-Frobenius ring (see [5, Ch. 13] for various characterizations of these rings). Therefore, left or right –modules of finite length are precisely the finitely generated ones. Over the Quasi-Frobenius ring , for every finitely generated right –module and for every finitely generated left –module (see [5, Theorem 13.3.2]).
If there is a non degenerate –bilinear form , then .
The following computation, which uses that both and are injective maps, gives the statement.
Assume that . An –bilinear form is left non degenerate if and only if it is right non degenerate. In such a case, both and are isomorphisms.
Assume that is right non degenerate, that is, the homomorphism of left –modules is injective. Since is QF, . Therefore, is an isomorphism. In order to prove that is left non degenerate, let such that . This means that, for every , . Since is surjective, we get that for all . This implies that , since is an injective cogenerator right –module. Therefore, is injective. We so far have proved that if is right non degenerate, then it is left non degenerate, and is an isomorphism. By symmetry, we get the full statement of the lemma. ∎
Now, for an –bilinear form , and subsets , , we define
Clearly, is a left –submodule of , while is a right –submodule of .
Assume that is non degenerate. For every –submodule (resp. ) of (resp. of ) we have that and .
From Lemma 2, . Consider the commutative diagram of left –modules
where for all . We see that is injective, so, being an isomorphism by Lemma 3, we get that is an isomorphism, too. This implies that is an isomorphism. Therefore, and as –modules. Analogously, we have isomorphisms of –modules and .
Now, observe that . Thus,
which is only possible if . Analogously, . ∎
With the notation of Proposition 4, we get from its proof that there are isomorphisms of –modules and , as well as and .
The map gives an anti-isomorphism, with inverse , between the lattices of right –submodules of and of left –submodules of .
If is associative, then the map gives an anti-isomorphism, with inverse , between the lattices of –bisubmodules of and of –bisubmodules of .
3. Non projective Frobenius algebras
Let be an algebra over a commutative Frobenius ring . Setting and in the framework of Section 2, we may thus consider the bimodules and . Therefore, is endowed with the right and left –module structures
This gives, indeed, an –bimodule structure on . Next, we materialize the discussion of Section 2 to this framework. Recall that a –bilinear form is associative if for all .
Let be an algebra over a commutative Frobenius ring . Assume to be finitely generated as a –module. The following structures related to are in bijective correspondence.
Associative non degenerate –bilinear forms
Isomorphisms of right –modules .
Isomorphisms of left –modules .
–linear forms such that .
–linear forms such that .
The equivalence between (1), (2) and (3) follows from Lemma 3 and the discussion previous to Definition 1. We only discuss the equivalence between (2) and (4), since that between (3) and (5) is symmetric. If there is an isomorphism of right –modules , then generates as a right –module. Conversely, given such that , we have the surjective homomorphism of right –modules given by . Since is Frobenius, , and we get that is an isomorphism, indeed. Let us finally argue that the linear form appearing in (4) coincides with that of (5). Let . Then . Thus, . ∎
An algebra over a commutative Frobenius ring is said to be non projective Frobenius if is finitely generated as a –module and there exits a non degenerate –bilinear form . The –linear form provided by Proposition 8 will be referred to as the Frobenius functional. It follows from Proposition 8 that a Frobenius –algebra in the sense of  is non projective Frobenius. The converse is not true (see Remark 19).
We have already seen that the additional structure maps that, according to Proposition 8, make a –algebra non projective Frobenius, are related by the following equalities
for all .
Every semisimple algebra over a commutative Frobenius ring with finitely generated is a non projective Frobenius algebra. By virtue of Wedderburn-Artin Theorem, it suffices by proving this when is simple. In this case, for a simple left –module . Now, as right –modules. Since is Frobenius, and have the same length as –modules. Therefore, and have the same length as –modules. This implies (look at as a matrix ring over a division –algebra) that is a simple right –module. Since its multiplicity in is , we deduce that as right –modules.
4. Annihilators in non projective Frobenius extensions
Let us come back to the situation of a pair of bimodules and with a non degenerate –bilinear map . Assume, in addition, that is a non projective Frobenius algebra over a Frobenius commutative ring , with Frobenius functional . The corresponding associative non degenerate –bilinear form will be denoted by , and it obeys the rule
Define, for each subset ,
which is a –submodule of . Analogously, for every subset we get the –submodule of
Indeed, and are the left and right orthogonal –submodules of and with respect to the –bilinear form
Obviously, and .
Let and be –submodules of and , respectively. Then
Thus, and are –submodules of, respectively, and . Moreover,
Let . If , then, since is a submodule of , we have
Now is arbitrary and is non degenerate, we thus get that . For the proof of the second part, observe that the bilinear form is non degenerate: if is such that for all , then . Hence, is left non degenerate. The argument on the right is similar. By Proposition 4, and . ∎
For a subset , we have the left (resp. right) ideal of defined by
For any non projective Frobenius algebra with Frobenius functional , every right ideal of , and every left ideal of , we have
Setting in Proposition 13, and the bilinear form to be the multiplication map of , we get the statement, since , while . ∎
It follows from Corollary 14 that every non projective Frobenius algebra over a Frobenius commutative ring is a Quasi-Frobenius ring. Next section is devoted to sharpen this result.
5. Non projective Frobenius algebras and Frobenius rings
Let an algebra over a commutative Frobenius ring with finitely generated. Let denote the Jacobson radical of . The socle of a right (or left) –module is the sum of all its simple –submodules. It is well known that, if is finitely generated, then . Therefore, may be computed as
where the last isomorphism of –modules (and, hence, of –modules) holds because is semisimple, and Example 12 applies.
Let be an algebra over a commutative Frobenius ring with finitely generated. Then is a non projective Frobenius –algebra if and only if is a Frobenius ring.
Assume first that is Frobenius as a ring. Then . By (1), . Now, is injective because it is Frobenius, and is injective because is Frobenius. Since they have isomorphic essential socles, it follows that , and is a non projective Frobenius algebra. Conversely, Corollary 14 implies that is Quasi-Frobenius. Since is non projective Frobenius, we have an isomorphism of –modules . Therefore, taking (1) into account, we get
Hence, is a Frobenius ring. ∎
Let be a finite ring of characteristic . Then is a Frobenius ring if and only if is a non projective Frobenius –algebra.
A non projective Frobenius algebra needs not to be projective over its commutative base Frobenius ring. One of the simplest examples is the ring , which is clearly Frobenius and, by Corollary 18, a non projective Frobenius algebra over . However, is not projective as a –module, so it is not a Frobenius algebra in the classical setting.
Next, we will describe a method for constructing Frobenius rings from skew polynomial rings with coefficients in a non projective Frobenius algebra.
Let be a non projective Frobenius algebra over a Frobenius commutative ring with Frobenius functional , and associative non degenerate –bilinear form
Consider the skew polynomial ring , where is a –algebra automorphism of , and let be a monic twosided polynomial, that is, . Since is a twosided ideal of , we get the –algebra , which is finitely generated as a –module because is monic. Indeed, every can be written as , for suitable with of degree smaller than . This implies, always with monic, that . In other words, we will identify the elements of with polynomials in with degree less than , with the operations made modulo . For , the notation stands for its term of degree . Finally, we assume that is a unit of .
is a non projective Frobenius -algebra with nondegenerate associative bilinear form
Therefore, is a Frobenius ring.
A straightforward computation modulo shows that, for , we have
Since is non degenerate, we deduce that . Now, let . Setting in (2), we have
Now, is arbitrary, is a unit, and is an automorphism of , so, we get from the non degeneracy of that . Thus, , and is non degenerate. By Theorem 16, is a Frobenius ring. ∎
6. Codes with a Frobenius alphabet
Let be a finite ring of characteristic , and consider as the group of –th roots of unit. Recall that the abelian group has a unique structure of ring, and that it is Frobenius. Now, we have
Let be a finite Frobenius ring of characteristic , and Frobenius functional (or generating characher) . Let a non degenerated bilinear form, where and are finite –modules. We know that and, by Lemma 3, and . Let see that this framework cover the module-theoretical setting considered in .
Consider an anti-automorphism , and a left module . We can consider the right –module whose underlying additive group is , with the right –module structure defined by for all . Then, as already observed in [10, Remark 4.9] a non degenerate –bilinear form is, precisely, a non degenerate sesquilinear form in the sense of [10, §3]. When has to be considered as the word ambient space for –linear codes, a canonical choice is to put with its canonical left –module structure.
Of course, the same finite abelian group may support both a left –module structure and a right –module structure. This case, considered in [10, §4], is clearly covered by our general formalism. Here, a natural choice is with its canonical –bimodule structure.
Next, we will see how the results in  are derived from our general theory. So, fix a non degenerate –bilinear form , where is a finite Frobenius ring of characteristic , and , are finite modules. Let be a Frobenius functional (that, is, a generating character). According to examples 21 and 22, this setting covers all cases considered in .
[10, Proposition 3.7 and Proposition 4.7] If is a left –submodule and is a right –submodule, then
By we denote the cardinal of a finite set .
[10, Theorem 3.6 and Theorem 4.6] Let and be submodules. Then and .
Recall that the Hamming weight of a vector is defined by the number of nonzero components of . Given an additive code , the Hamming weight enumerator of is the complex polynomial in two variables
A general version of McWilliams identity appears in [10, Theorem 5.2]. Unfortunately, it is not valid for every non degenerate bilinear form, as the following example shows.
Set , and let the non degenerate –bilinear form defined by the non singular matrix
that is, for all . Let the linear code . Its dual (with respect to ) is . Therefore,
If the McWilliams identity stated in [10, Theorem 5.2] was applicable to , then we would have
Therefore, the identities from [10, Theorem 5.2] do not hold for a general non degenerate bilinear form.
are isometries with respect to the Hamming weights in and . This is the statement of [10, Lemma 5.3], which does not hold for an arbitrary non-degenerate –bilinear form on . The argument is valid, however, for suitable bilinear forms. To be more precise, let denote, for , the vector of whose only nonzero component is the –th, which is .
[10, Theorem 5.2] Let a non degenerate –bilinear form such that there is a generating character on such that for all . Let be a left (resp. right) –linear code , and set (resp. ). Then
We conclude this section by giving some examples of non projective Frobenius algebras for which the annihilators with respect to their associative bilinear forms are the euclidean duals with respect to certain bases. The notation will be used to denote the Euclidean dual of a given code , with respect to the Euclidean product which will be clear in each situation.
Let be a finite group, and consider the group algebra , which is a basic example of Frobenius –algebra, with the associative nondegenerate bilinear form defined by
Note that, if we denote by the obvious euclidean bilinear form on , then , for all , where denotes the involution determined by for . Hence, for every subset of , which implies that if is a left ideal of , then is a left ideal of , too.
Let be an automorphism of a finite Frobenius ring of characteristic . Consider a Frobenius functional (or generating character) , and the non degenerate –bilinear form given by for all . Let be a multiple of the order of . The polynomial is central, so we may consider the finite ring , which is a Frobenius ring and a non projective Frobenius –algebra, according to Proposition 20, with the nondegenerate associative –bilinear form
On the other hand, looking at as free –module of rank with basis , we have the Euclidean –bilinear form
Let be defined by
Then is a –algebra involution. Moreover, for every left –submodule of , we have
which implies the equality . ∎
If is a –cyclic code, then is a –cyclic code.
Since is a left ideal of , we get that is a right ideal of and, therefore, is a left ideal. By Proposition 28, becomes a left ideal of . ∎
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