Exploring the multiplicative structure of Z/nZ

The set of integers modulo n, for a fixed integer n, is well known as a ring also as an additive group. In this thesis, we focus exclusively on its structure as a multiplicative monoid, when endowed with the multiplication only. To do this, our approach relies on the study of monoid homomorphisms. In particular, we investigate the homomorphisms between two prime power multiplicative modular monoids, aiming to understand their algebraic and structural properties. These sets of homomorphisms provide valuable insight into the actions, semidirect products, endomorphisms, and automorphisms associated with such monoids. Considering the set of homomorphisms of prime power multiplicative modular monoids, with the law operator of the codomain, we identify its algebraic structure as a monoid. We also determine the algebraic structure of the Aut(Z/peZ, ⋅), where p is a prime and e ∈ N as a group when endowed with composition. From the observation that any automorphism in Aut(Z/peZ, ⋅) induces an automorphisms in Aut(Upe , ⋅), we asked ourselves which of the automorphisms of (Upe , ⋅) can be lifted to an automorphism of (Z/peZ, ⋅). The answer we found is that they are precisely those in Aut(Upe , ⋅) that induce an automorphism of (Upf , ⋅) for all f ≤ e. We denote the set of such automorphisms by Ape . With this important result, we are able to prove that Aut(Z/peZ, ⋅) is the semidirect product of (Upe−1 , ⋅) and (Ape , ○). It now remains to understand the group structure of Ape endowed with composition. We prove that (Ape , ○) is always an abelian group independently of the parity of p. Along the way, we prove that for e ≥ 4, (Aut(U2e , ⋅), ○) is the direct product of Z/2Z with the central product of a dihedral group of order 8 and the cyclic group Z/2e−3Z. We next address the general case. Computational experiments (implemented in Python) show that, for small prime powers, the automorphism group of a product of distinct prime-power multiplicative modular monoids splits as the product of the automorphism groups of the individual factors. Building on this evidence, we prove a significantly stronger statement. For a finite family of pairwise distinct finite D-rings that are total rings of fractions, the automorphism group of the product of their underlying multiplicative monoids decomposes as the product of the automorphism groups of the components. The proof rests on two ideas. First, endomorphisms of a finite product of monoids admit a matrix-like description, much as in the group case; this makes it possible to isolate how different factors can (or cannot) interact. Second, a key lemma shows that any monoid homomorphism from a ring to a D-ring is either trivial or necessarily sends the zero element to the zero element. Together, these ingredients force automorphisms to act componentwise, yielding the desired product decomposition.

Grantor dc:publisher

Stellenbosch : Stellenbosch University

Year dc:date.issued

2026

Author dc:creator

• Atalaye, Joseph

Advisors dc:contributor.advisor

• Marques, S.
• Baker, L.

Language dc:language.iso

en

Repository record dc:identifier.uri

https://scholar.sun.ac.za/handle/10019.1/135563

OAI identifier oai:identifier

oai:scholar.sun.ac.za:10019.1/135563