Finding algebraic conditions on a Banach algebra which lead to finite dimensionality is a fairly old problem in the Banach algebra context. For concrete Banach algebras which are associated with a locally compact group or a discrete semigroup, this problem is of special interest and there is already a vast literature on it. The paper under review is one of this kind, with focus on the finite-generation condition. The author was motivated by Dales-Żelazko’s conjecture which asserts that a unital Banach algebra \(\mathcal{A}\) is finite dimensional whenever all of its maximal left ideals are finitely-generated [H. G. Dales and W. Żelazko, Stud. Math. 212, No. 2, 173–193 (2012; Zbl 1269.46028)].
A left ideal \(\mathbb I\) in a Banach algebra \(\mathcal{A}\) is finitely-generated if there is a finite subset \(E\) of \(\mathcal{A}\) such that \(\mathcal{I}\) is equal to the linear span of \(\mathcal{A}^{\#} E\) where \(\mathcal{A}^{\#}\) is the unitization of \(\mathcal{A}\). Let \(G\) be a locally compact group with left Haar measure \(m\) and \(\omega\) be a weight on \(G\), that is, a submultiplicative function from \(G\) into \([1,\infty)\) with \(\omega(e)=1\). The augmentation ideal \(L_0^1(G,\omega)\) [resp., \(M_0(G,\omega)\)] of the weighted group algebra \(L^1(G,\omega)\) [resp., measure algebra \(M(G,\omega)\)] is the set of all \(f\in L^1(G,\omega)\) [resp., \(\mu\in M(G,\omega)\)] for which \(\int fdm=0\) [resp., \(\mu(G)=0\)]. When \(\omega\) is the constant weight, in all of these notations the word \(\omega\) is dropped. For discrete monoids, these concepts are defined analogously. In the three cases of non-discrete groups, discrete monoids and discrete semigroups, finitely-generated augmentation ideals of weighted group/semigroup algebras for certain weights are identified. The case of non-discrete groups is considered first. It is shown that \(M_0(G)\) is finitely-generated if and only if \(G\) is compact. As a consequence, \(L_0^1(G)\) is finitely-generated if and only if \(G\) is finite. Then the conjecture of Dales-Żelazko is verified for \(M(G,\omega)\) when \(G\) is not discrete. Let \(M\) be a discrete monoid. A necessary and sufficient condition is given for the augmentation ideal \(\ell^1_0(M)\) of \(\ell^1(M)\) to be finitely-generated. This later condition is equivalent to finiteness if \(M\) is weakly right cancellative, that is, for every \(a,b\in M\), the set \(ab^{-1}=\{ x\in M\;:\;xb=a\}\) is finite. Consequently, Dales-Żelazko’s conjecture for \(\ell^1(M)\) has a positive answer, whenever \(M\) is a weakly right cancellative monoid. For certain weights on a discrete, finitely generated group \(G\), it is shown that \(\ell^1_0(G,\omega)\) is finitely-generated. Finally, those weights on \(\mathbb Z\) for which the augmentation ideal of \(\ell^1(\mathbb Z,\omega)\) is finitely-generated, are characterized. Some of the cases where Dales-Żelazko’s conjecture remains open, are also pointed out for future research.