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The first aim of this work is to present the main results and methods of the theory of Noetherian semigroup algebras. These general results are then applied and illustrated in the context of certain interesting and important concrete classes of algebras that arise in a variety of areas and have been recently intensively studied.

This chapter is devoted to the necessary background on semigroups and especially on semigroups of matrices over a division algebra, referred to as linear semigroups.

This chapter is devoted to the necessary background on ring theory. Our investigations force us to consider a rather wide variety of topics. Only results essential for later use are stated. For the convenience of the reader we mostly will refer to standard references.

As explained in Section 3.3, group algebras of polycyclic-by-finite groups are the only known examples of Noetherian group algebras. In the search for more classes of Noetherian rings, the first obvious step is to investigate semigroup algebras that are subalgebras of Noetherian group algebras.

In this chapter we prove certain fundamental general results on right Noetherian semigroup algebras K[S] . First, we show that in many important cases such algebras are finitely generated. In particular, this extends the observation made in Theorem 4.1.7 for submonoids of polycyclic-by-finite groups.

In this chapter we study semigroup algebras K[S] that are principal right ideal rings. First, we show that these are finitely generated PI algebras of Gelfand-Kirillov dimension at most 1.

It remains an unsolved problem to characterize when an arbitrary semigroup algebra K[S] over a field K is a prime Noetherian maximal order. In this chapter we describe when a semigroup algebra K[S] is a Noetherian PI domain which is a maximal order. This result will be applied in the context of concrete classes of algebras considered in Chapter 8.

As promised in Chapter 4, we investigate in this chapter an important class of Noetherian semigroup algebras of monoids S that arise in other contexts. The monoids considered are called monoids of (left) I -type and they were introduced by Gateva-Ivanova and Van den Bergh. Their work was inspired by earlier work of Tate and Van den Bergh on Sklyanin algebras.

In this chapter, we investigate finitely presented monoids S , say with generating set {x_1, ⋯, x_n}, that have square free quadratic homogeneous defining relations, that is, the relations are of the form x_ix_j = x_kx_l, where i ≠ j and k ≠ l.

The theory developed in the preceding chapters will be illustrated on several examples of monoids of skew type and their algebras. Our calculations also show that several ingredients of the structural description of these combinatorially defined algebras can be effectively computed.