Invertible Algebras with an augmentation map

We briefly survey results about Invertible Algebras (algebras having bases that consist entirely of units) and other related notions. In addition, we consider the existence of an augmentation map as a possible way in which results about group rings, the archetypical invertible algebras, may be extended to more general settings. 1 Invertible Algebras and Some Related Notions Section 1 of this paper surveys the study of Invertible Algebras (over not-necessarily commutative rings) and other related notions. Section 2 introduces new ideas analogous to the augmentation map of group rings. Invertible algebras are those algebras that satisfy the condition that they have a basis consisting entirely of units. Our brief survey also touches on a few other related concepts. The concept of invertible algebras was originally introduced in [10] and is the subject of two recent papers [8] and [11]. The somewhat related notion of fluidity is studied in the upcoming paper [9]. Other notions mentioned in this first section are k-good rings (c.f. [17]) and S-rings (c.f. [15]). Group Rings are clearly an example of invertible algebras; their theory is very well developed and is central in many areas of mathematics. Standard references for group rings include the classics [13] and [14]. Section 2 points out a direction of research aiming to extend results about group rings to a larger family of invertible algebras. This is done by characterizing precisely those algebras that have an augmentation map. In this paper, when we use the expression A is an R-algebra we deviate from the standard use of that terminology in two ways: one which narrows the net that we cast and another one that widens it. First, we do not allow a proper homomorphic image of the ring R to be contained in A; according to the definition we use in this paper, R itself is contained in A. The second difference is that R is not necessarily assumed to be contained in the center of A; in fact, we do not even assume that the ring R is commutative. Also, a feature that will be common to all those algebras considered here is that they will be free as (left-) R-modules. In other words, our setting is that we have a ring A that has a subring R such that A is a free left R-module. Definition 1.1. LetA be an algebra over a ringR and B be a basis forA overR. B is an invertible basis if each element of B is invertible in A. If B is an invertible basis such that B−1, the set of the inverses of the elements of B, also constitutes a basis then B is an invertible-2(I2) basis. An algebra with an invertible basis is an invertible algebra and an algebra with an I2 basis is an I2 algebra. Various papers in the literature have considered properties of rings having to do with expressing their elements in terms of sums of units. See ([4], [16] and [18]), for example. In particular, S-rings and k-good are defined in ([15] and [17]) as follows: Definition 1.2. In [17], for k ∈ Z+, Vámos calls a ring R a k-good ring if each element of R is the sum of k units. A related concept is that of an S-ring, a ring in which every element is the sum of units. (c.f. [15]). Lemma 1.3 (Lemma 1, [15]). Let A be a ring, let G be a group and let R be the group ring defined by A and G. Then R is an S-ring if and only if A is an S-ring. Lemma 1.4 (Lemma 5, [15]). (a) In an even S-ring, zero can be written as the sum of an odd number of units. (b) If R is an S-ring that contains a unit u such that u+ 1 is a unit, then R is an even S-ring. Invertible Algebras with an augmentation map 367 (c) A finite product of even S-rings is an even S-ring. (d) If R is any ring, then Rn is an even S-ring for all n > 1. (e) If R is an even S-ring, and S is an S-ring, then R⊕ S is an S-ring. Proposition 1.5 (Proposition 6, [15]). Let R be completely reducible. Then R is an S-ring if and only if the two-element field occurs at most once in the decomposition of R into completely reducible simple rings. R is an even S-ring if and only if this field does not occur at all. As usual, we denote the ring of n×nmatrices over an arbitrary ringR byMn(R). Performing an elementary row operation on the identity matrix results in an elementary matrix. Definition 1.6. In [17], the author defines the elementary group of Mn(R), denoted by En(R), as the subgroup of GLn(R) generated by elementary matrices, permutation matrices and -1. Definition 1.7. A matrix is strongly k-good if it can be written as a sum of k elements of the elementary group, and a matrix ring is strongly k-good if every matrix is strongly k-good (see [17].) Lemma 1.8 (Lemma 5, [17]). Let R be an arbitrary ring and n ≥ 2. Then any diagonal matrix in Mn(R) is strongly 2-good. Proposition 1.9 (Proposition 6, [17]). A proper matrix ring over an elementary divisor ring is 2-good. Over an Euclidean domain proper matrix rings are strongly 2-good. Proposition 1.10 (Proposition 8, [17]). Let R be a ring, m,n ≥ 1 and k ≥ 2. If the matrix rings Mn(R) and Mm(R) are both k-good, then so is the matrix ring Mn+m(R). Definition 1.11. Let R be a subring of S. R is said to be unit closed in S if no nonunit of R becomes a unit in S, i.e. U(S) ∩R = U(R) (see [17].) Proposition 1.12 (Theorem 13, [17]). Every ring can be embedded as a unit closed subring in a 2-good ring. The study of k-good rings has been furthered recently in [5] which yielded the following interesting results: Proposition 1.13 (Theorem 1, [5]). For a right self-injective ringR, the following conditions are equivalent: (1) Every element of R is a sum of two units. (2) Identity of R is a sum of two units. (3) R has no factor ring isomorphic to Z2. Proposition 1.14 (Theorem 3, [5]). Let MS be a quasi-continuous module with finite exchange property and R = EndS(M). Then every element of R is a sum of two units if and only if no factor ring of R is isomorphic to Z2. Proposition 1.15 (Proposition 7, [5]). If R is a right self-injective ring and G a locally finite group, then every element of RG is a sum of two units unless R has a factor ring isomorphic to Z2. The study of invertible-2 algebras naturally leads to considering questions about when the sets of inverses of linearly independent sets of units are linearly independent. This condition seems hard to satisfy and indeed the phenomena described here seem rather rare. We start by introducing appropriate terminology from [9]. Definition 1.16. (i) A linearly independent set S of units of the algebra A is said to be fluid if S−1, the set of inverses of its elements, is also linearly independent. (ii) An algebra A is said to be fluid if every linearly independent set of units S of A is fluid. (iii) In order to prevent vacuous nonsensical consequences, we must introduce the following parameter: for an algebra A over a ring R, the mojo of A (mojo(A)) is the largest number of linearly independent units one can find in A. Clearly, if A is free as a module over R then mojo(A) ≤ rankR(A). When A has finite rank as a free module over R then mojo(A) = rankR(A) if and only if A is an invertible algebra. 368 Sergio R. López-Permouth and Jeremy Moore (iv) For a number t ≤ mojo(A), we say that the algebraA is t-fluid if every linearly independent set of units S with at most t elements is fluid. The fluidity of A (fluid(A)) is the largest t such that A is t-fluid. Clearly if fluid(A) = rankR(A) then A is invertible-2. We start with a few immediate remarks. Remark 1.17. (i) Every linearly independent set of units S having exactly two elements is fluid. (ii) If mojo(A) ≥ 2 then fluid(A) ≥ 2. (iii) A = F [x] is an example of an F -algebra with mojo(A) = fluid(A) = 1. (iv) If A is a subalgebra of B then if B is fluid then so is A. The following example shows that it is indeed possible for the fluidity, mojo and dimension of an algebra to be different from one another. Example 1.18. LetA = T3(F2), the ring of upper triangular matrices over F2. Clearly dim(A) = 6. It is straightforward to show 4 ≤ mojo(A) 6= 6. Furthermore, we easily find a set of 4 linearly independent units whose inverses are linearly dependent, thus fluid(A) < 4. Fluid fields extensions are rather scarce, in fact, that is the subject of the following proposition from [9]. Proposition 1.19. Let E be a field extension over F. Then fluid(E) = 2 and therefore, E/F is fluid if and only if the degree of E over F is 2. As we will show in Proposition 1.27, matrix algebras have played an important role in the study of invertible algebras. It is therefore also interesting to consider their fluidity. The next proposition from [9] announces that, in general, in spite of being I2, matrix algebras are far from being fluid and, in fact their fluidity is almost always 2. Proposition 1.20. Let R be a commutative ring such that 1 = a+ b where a and b are units and n ≥ 3. Then the fluidity of Mn(R) is 2. The technical condition that the identity of R be a sum of two units is not necessary for the result to hold. This is illustrated in the following proposition. In fact, while we do not know any way to remove the hypothesis from Proposition 1.20, we also do not know at this moment any examples where the result fails. Proposition 1.21. Let R = F2 and n ≥ 3. Then the fluidity of Mn(R) is 2. One of the first remarkable results about invertible algebras is the following characterization of group rings from [10]. Proposition 1.22 (Proposition 2.12, [10]). Let A be an algebra over a ring R. A is a group ring if and only if A has an invertible basis that is closed under products and whose elements commute with those of R. Proposition 1.22 has as a corollary which strengthens a result about field extensions reported in [6] for reals over rationals and in general in [12]. Namely, Corollary 1.23 extends the result that no proper field extension has a basis that is closed under multiplication. Corollary 1.23. If a simple ring A is an invertible R-algebra with invertible basis B 6= 1 then B is not closed under products. Propositon 1.22 was later generalized in [8] to include characterizations of the various crossed products as algebras having invertible bases with additional properties which are softer versions of those in Proposition 1.22. Definition 1.24. Let A be an algebra over a ring R, and B an R-basis for A. If for all v ∈ B there exists σv ∈ Aut(R) such that for all r ∈ R, vr = σv(r)v then R scalarly commutes with B. In the case that for all v ∈ B, σv = 1 we naturally say R commutes with B. Definition 1.25. Let A be an algebra over a ring R and B be an invertible basis for A over R. If for all v, w ∈ B, αvw ∈ B for some α ∈ U(R) then B is scalarly closed under products. Proposition 1.26 (Proposition 2.7, [8]). Let A be an algebra over a ring R. Invertible Algebras with an augmentation map 369 (i) A is a crossed product if and only if A has an invertible basis that is scalarly closed under products and whose elements scalarly commute with those of R. (ii) A is a skew group ring if and only if A has an invertible basis that is closed under products and whose elements scalarly commute with those of R. (iii) A is a twisted group ring if and only if A has an invertible basis that is scalarly closed under products and whose elements commute with those of R. (iv) A is a group ring if and only if A has an invertible basis that is closed under products and whose elements commute with those of R. In [10] it was shown that over any ring R, the matrix ring A = Mn(R) (for any n ≥ 1) is an I2 R-algebra. Then the result was significantly extended in [8]. Proposition 1.27 (Proposition 3.1, [8]). Let A be an algebra over a ring R with a basis that includes 1. Then Mn(A) is an invertible algebra over R for all n ≥ 2. Corollary 1.28. Invertibility is not a Morita invariant. Proposition 1.29. Let D be a division ring and let A be a semilocal D-algebra. If D 6= F2 then A is invertible. If D = F2 then A is invertible if and only if A does not admit an algebra epimorphism to F2 ⊕ F2. Proposition 1.30 (Proposition 4.6, [8]). Let R be a ring and let A be a free local R-algebra. Then A is invertible. 2 Toward an analogue of the augmentation map Group rings were one of the original motivations for our study of invertible algebras. As it has been shown in the previous sections, the class of invertible algebras is much bigger than that. It is therefore not to be expected that many of the results concerning structural properties of group rings can be extended to invertible algebras. We investigate, however, conditions that may allow some of those results to extend. Our first observation is that the idea of an augmentation map can sometimes be extended to a more general setting. We start by describing properties that will allow the augmentation map φ : A→ R defined by φ( ∑ i αivi) = ∑ i αi, to become a ring homomorphism and we will show that these conditions are indeed necessary and sufficient in Proposition 2.8. It is well-known that for a finite group G, the group ring R[G] is self-injective if and only if R is self-injective [3]. The next example illustrates how this result does not extend to invertible algebras. Example 2.1. It is well known that a finite-dimensional commutative local algebra A over a field R = F , is self-injective if and only if A has a unique minimal ideal [7]. Consider the F3-algebra A = F3[x,y] 〈x,y〉2 . It is easily checked that the basis B = {1, 1 + x, 1 + y} is an invertible basis for A. Now 〈x, y〉 is the unique maximal ideal of A and so A is a local ring. Now 〈x〉, and 〈y〉 are both minimal ideals and therefore A does not have a unique minimal ideal. Therefore, A is not self-injective even though it is finite dimensional over F (which, as all fields, is self-injective.) A key element of the proof in [3] of the above characterization of self-injective group rings is the fact that the R-homomorphism φ : R[G] → R given by φ( ∑ g∈G αgg) = ∑ g∈G αg is a ring homomorphism. It seems reasonable to ask, for a basis B of an algebra A over a ring R, when the map φ : A → R given by φ( ∑ b∈B αbb) = ∑ b∈B αb is a ring homomorphism. We introduce next a few definitions that will be essential components of the answer to that question provided by Proposition 2.8 below. Furthermore, these definitions and Proposition 2.8 will be instrumental in providing a partial converse to Proposition 2.2 in [10]. Definition 2.2. Let A be an R-algebra with basis B. We say that R commutes linearly with B if for all v ∈ B and β ∈ R, if vβ = ∑ vk∈B δkvk then ∑ δk = β. Definition 2.3. Let A be an R-algebra with basis B. We call B linearly closed under products if for all v, w ∈ B, if vw = ∑ vi∈B αivi then ∑ vi∈B αi = 1. Definition 2.4. Let A be an R-algebra with invertible basis B. We say B is linearly closed under inverses if for all v ∈ B, if v−1 = ∑ vi∈B αivi then ∑ vi∈B αi = 1. 370 Sergio R. López-Permouth and Jeremy Moore Clearly Definition 2.3 is satisfied by group rings. However, in [10], an example of an invertible algebra that is not a group ring is given, namely F2[x, y] 〈x, y〉2 . An invertible basis of this algebra also illustrates that Definition 2.3 does not just pertain to group rings. Example 2.5. ConsiderA = F2[x, y] 〈x, y〉2 . As stated in [10] we knowA is not a group ring. However, A is an invertible algebra with invertible basis B = {1, 1 + x, 1 + y}. The product of 1 + x and 1 + y is (1 + x)(1 + y) = 1 + x+ y = 1(1) + 1(1 + x) + 1(1 + y). The sum of the coefficients is 1. The other combinations are trivial. Therefore, B satisfies Definition 2.3. Proposition 2.6. Let A be an invertible R-algebra with invertible basis B. Assume B is linearly closed under products and scalarly closed under products. Then B is a group. Proof. Let vi, vj ∈ B. Since B is scalarly closed under products we have vivj = αvk. But since B is linearly closed under products we must have α = 1. Therefore, B is closed under products and by Proposition 2.12 in [10], B is a group. An obvious question is are there other examples of algebras with bases that are linearly closed under products and inverses, yet are not group rings. The previous example inspired a consideration of algebras of the form R[x1, x2, . . . , xn] 〈x1, x2, . . . , xn〉 . The following proposition will show there are numerous examples of algebras that are linearly closed under products and inverses yet not group rings. Proposition 2.7. Let A = R[x1, x2, . . . , xn] 〈x1, x2, . . . , xn〉 where R is any ring, n ≥ 1 and m ≥ 2. Assume {xi|i = 1, . . . n} is a commutative set. Then A has an invertible basis that is linearly closed under products and inverses, namely, B = {1} ⋃ {1 + x1 1 x r2 2 · · ·xn n } where 0 ≤ ri < m for all i and 1 ≤ n ∑