가환환 상의 모노이드에 대한 꼬임 함수 연구

Abstract

Let 𝑅 be a commutative ring with identity, 𝑈(𝑅) be the set of units of 𝑅, 𝛤 be a nonzero commutative additive monoid, and 𝑡 be a twist function of 𝛤 on 𝑅. Passman introduced the twisted semigroup ring 𝑅 ͭͭ[𝑋;𝛤] of 𝛤 on 𝑅 with respect to the twist function 𝑡, and studied the algebraic properties on such a ring. It is well known that the twisted semigroup ring 𝑅 ͭͭ[𝑋;𝛤] becomes a commutative ring with identity. Let 𝑇(𝑅;𝛤) be the set of twist functions of 𝛤 on 𝑅. We are interested in twist functions of 𝛤 on 𝑅 instead of twisted semigroup rings. In this thesis, we introduce several examples of twist functions, and study the properties of twist functions of 𝛤 on 𝑅. More precisely, we show that the set of twist functions of 𝛤 on 𝑅 forms a group. If ℕ is the set of nonnegative integers and ℤ is the ring of integers, then we also show that the set of twist functions of ℕ on ℤ forms a Boolean group.