### Abstract

Given an action α of a monoid T on a ring A by ring endomorphisms, and an Ore subset S of T, a general construction of a fractional skew monoid ring Sop*αA* αT is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case S is a subsemigroup of a group G such that G=S-1S, we obtain a G-graded ring Sop*αA*αS with the property that, for each s ε S, the s-component contains a left invertible element and the s-1-component contains a right invertible element. In the most basic case, where G=ℤ and S=T=ℤ+, the construction is fully determined by a single ring endomorphism α of A. If α is an isomorphism onto a proper corner pAp, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by A[t+,t-;α]. Examples of this construction are given, and it is proven that several classes of known algebras, including the Leavitt algebras of type (1,n), can be presented in the form A[t+,t-;α]. Finally, mild and reasonably natural conditions are obtained under which Sop*αA* αS is a purely infinite simple ring. © 2004 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 104-126 |

Journal | Journal of Algebra |

Volume | 278 |

DOIs | |

Publication status | Published - 1 Aug 2004 |

### Keywords

- Leavitt algebra
- Purely infinite simple ring
- Skew monoid ring

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## Cite this

Ara, P., González-Barroso, M. A., Goodearl, K. R., & Pardo, E. (2004). Fractional skew monoid rings.

*Journal of Algebra*,*278*, 104-126. https://doi.org/10.1016/j.jalgebra.2004.03.009