Ring & Module Structure Theory

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Abstractions about the structural properties of rings and modules, covering finiteness and dimension conditions (finite algebras, Jaffard rings, length of a module), depth conditions (Cohen-Macaulay and Buchsbaum rings), gradings (graded rings, supermodules), special ideal types like test ideals, and decomposition results for PID and serial modules.

14 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Buchsbaum ring — A Noetherian local ring in which every system of parameters is a weak sequence under a maximal-ideal colon condition.
• Characteristic (algebra) — The characteristic of a ring is the least positive integer n for which n copies of the multiplicative identity sum to zero, or zero when no such integer exists.
• Cohen–Macaulay Ring — A commutative Noetherian ring whose localizations all have depth equal to Krull dimension.
• Dualizing module — In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety.
• Finite algebra — An R-algebra that is finitely generated as an R-module, a stronger finiteness condition than finite generation as an algebra.
• Graded Ring — A ring decomposed into additive homogeneous components whose products have the sum of their component degrees.
• Idealizer — In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal.
• Jaffard ring — In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
• Length of a module — In algebra, the length of a module over a ring R is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules.
• Regular ideal — In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ex - x \in \mathfrak{i} for every x \in A .
• Serial Module — A module admitting a direct-sum decomposition into uniserial modules, each with submodules linearly ordered by inclusion.
• Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain — A classification theorem stating that every finitely generated module over a principal ideal domain decomposes into a finite free part and uniquely determined cyclic torsion factors, in invariant-factor or elementary-divisor form.
• Supermodule — In mathematics, a supermodule is a Z 2 -graded module over a superring or superalgebra.
• Test ideal — A test ideal is a positive characteristic analog of a multiplier ideal in, say, the field of complex numbers.