  In this paper we discuss the relationship between noninvertible topological
operators, one-form symmetries, and decomposition of two-dimensional quantum
field theories, focusing on two-dimensional orbifolds with and without discrete
torsion. As one component of our analysis, we study the ring of dimension-zero
operators in two-dimensional theories exhibiting decomposition. From a
commutative algebra perspective, the rings are naturally associated to a finite
number of points, one point for each universe in the decomposition. Each
universe is canonically associated to a representation, which defines a
projector, an idempotent in the ring of dimension-zero operators. We discuss
how bulk Wilson lines act as defects bridging universes, and how Wilson lines
on boundaries of two-dimensional theories decompose, and compute actions of
projectors. We discuss one-form symmetries of the rings, and related
properties. We also give general formulas for projection operators, which
previously were computed on a case-by-case basis. Finally, we propose a
characterization of noninvertible higher-form symmetries in this context in
terms of representations. In that characterization, non-isomorphic universes
appearing in decomposition are associated with noninvertible one-form
symmetries.
