  Many quantitative approaches to the dynamical scrambling of information in
quantum systems involve the study of out-of-time-ordered correlators (OTOCs).
In this paper, we introduce an algebraic OTOC ($\mathcal{A}$-OTOC) that allows
us to study information scrambling of generalized quantum subsystems under
quantum channels. For closed quantum systems, this algebraic framework was
recently employed to unify quantum information-theoretic notions of operator
entanglement, coherence-generating power, and Loschmidt echo. The main focus of
this work is to provide a natural generalization of these techniques to open
quantum systems. We first show that, for unitary dynamics, the
$\mathcal{A}$-OTOC quantifies a generalized notion of information scrambling,
namely between a subalgebra of observables and its commutant. For open quantum
systems, on the other hand, we find a competition between the global
environmental decoherence and the local scrambling of information. We
illustrate this interplay by analytically studying various illustrative
examples of algebras and quantum channels. To complement our analytical
results, we perform numerical simulations of two paradigmatic systems: the PXP
model and the Heisenberg XXX model, under dephasing. Our numerical results
reveal connections with many-body scars and the stability of decoherence-free
subspaces.
