Associative algebras admitting a quasi-multiplicative basis.
A multiplicative basis of an (associative) algebra \(R\) over a field \(K\) is a basis \(B=\{e_i\}\), \(i\in J\), such that \(B\cup\{0\}\) is multiplicatively closed. Examples include \(M_n(K)\) and group algebras \(KG\) and, if \(K\) is algebraically closed, quiver algebras and algebras of finite representation type [\textit{Z. Bahlogh}, Math. Commun. 12, No. 2, 229-238 (2007; Zbl 1156.20003); \textit{R. Bautista} et al., Invent. Math. 81, 217-285 (1985; Zbl 0575.16012); \textit{V. Bovdi}, Arch. Math. 74, No. 2, 81-88 (2000; Zbl 0986.20004); \textit{H. Kupisch}, J. Reine Angew. Math. 201, 100-112 (1959; Zbl 0094.25101); ibid. 219, 1-25 (1965; Zbl 0132.28002); \textit{H. Kupisch} and \textit{J. Waschbüsch}, Math. Z. 186, 401-405 (1984; Zbl 0549.16013); \textit{A. V. Roiter} and \textit{V. V. Sergeichuk}, Ukr. Math. J. 46, No. 5, 604-617 (1994) and Ukr. Mat. Zh. 46, No. 5, 567-579 (1994; Zbl 0842.16009)]. Motivated by other examples of algebras [\textit{Yu. A. Bahturin} et al., Sb. Math. 199, No. 7, 965-983 (2008; Zbl 1258.16046); translation from Mat. Sb. 199, No. 7, 21-40 (2008); the author, Rep. Math. Phys. 69, No. 1, 75-86 (2012; Zbl 1263.16045); \textit{S. P. Conlon}, J. Aust. Math. Soc. 4, 152-173 (1964; Zbl 0134.26305); \textit{M. Kochetov}, Acta Appl. Math. 108, No. 1, 101-127 (2009; Zbl 1230.17025)] in the paper a more general class of (associative) algebras is introduced and studied, namely algebras \(R\) with a quasi-multiplicative basis (q.m.b.), defined by the following property \((V,B)\): There are \(K\)-subspaces \(W\neq 0\) and \(V\) of \(R\) and a basis \(B=\{e_i\}_{i\in J}\) of \(W\) such that (i) \(R=V\oplus W\), (ii) \(e_ie_j=Ke_t\) for some \(t\in J\) or \(e_ie_j\in V\), (iii) \(Ve_i\subset Ke_j\) and \(e_iV\subset Ke_t\) for some \(i,t\in J\), (iv) \(VV\subset Ke_i\) for some \(i\in J\) or \(VV\subset V\). The study is aimed at a decomposition of \(R\) with respect to \(V\) and \(B\). Central concept is an intricate equivalence relation among the elements of \(J\), the \textit{connection}, realized by the ``connection technics, the main tool in the study'' [c.f. ``support technics'', the author, J. Math. Phys. 50, No. 10, 103513 (2009; Zbl 1275.17043)]. The main results are: 1. To each equivalence class \([i]\), \(i\in J\), is associated an ideal \(A_{[i]}\) of \(R\) with a q.m.b. induced by \((V,B)\), i.e. \(A_{[i]}=V_{[i]}\oplus W_{[i]}\), where \(V_{[i]}\) and \(W_{[i]}\) are subspaces of \(V\) and \(W\), resp., \(W_{[i]}\neq 0\) and a subset \(B_{[i]}\) of \(B\) is a basis of \(W_{[i]}\). 2. \(R\) has a decomposition \[ R=U\oplus\Biggl(\sum_{[i]}A_{[i]}\Biggr),\;U\text{ a subspace of }V.\tag{*} \] 3. Under additional (``mild'') conditions (*) yields the direct decompositon \(R=\bigoplus_{[i]}A_{[i]}\), and the family of the \(A_{[i]}\) is precisely the family of all ideals \(A_j\neq 0\) of \(R\) which are minimal with respect to the existence of a q.m.b. induced by \((V,B)\).
scientific article; zbMATH DE number 6384245