Algebraic structure changes when raising to a power

While algebraic structures are the main objects of study of modern algebra,

they are present in many mathematical disciplines. This has not only been

a thriving force for the development of the field of algebra itself, it has

also led to additional perspectives in other fields by focusing on structural

similarities. It allows one to translate and adopt methods and tools from

different mathematical fields which are based on the given structure.

In this context, structural decompositions are typically used to characterize

objects of interest in terms of their “building blocks” as well as the reduction

of questions to potentially easier-to-handle cases. The factorization of

integers into prime powers, polynomials into irreducible factors, and the

decomposition of algebraic varieties into their irreducible components are

only some of many structural decompositions.

The objects of interest in this talk are elements and ideals of commutative

rings. Focus is set on the fact that the indecomposable building blocks of

an element or ideal are subject to change when raising it to a power.