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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
3
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The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings)
At least in commutative situations, I would argue that a good notion is simply an ideal in $H^0(R)$.
For example, the theory of local cohomology works just as well as it does for commutative rings, a …