Ring & Module Theory

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Abstractions about the internal structure of rings and modules, including classification by ideal and radical properties (radical of a ring, semiprimitive ring, primitive ring), homological invariants of modules (depth, grade, free presentation), and special ring types defined by structural conditions (perfect ring, polynomial identity ring, Gelfand ring).

17 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.

• Central simple algebra — A finite-dimensional associative algebra over a field that has no nontrivial two-sided ideals and whose center is exactly the base field.
• Depth (ring theory) — A homological invariant measuring the length of a maximal regular sequence acting on a module, equivalently the first degree of nonvanishing Ext under standard local Noetherian hypotheses.
• Different ideal — An ideal measuring the failure of the ring of integers of a number field to be self-dual under the trace pairing, inverse to the codifferent fractional ideal.
• Domain (ring theory) — A nonzero ring with no nonzero left or right zero divisors.
• Free presentation — An exact sequence of free modules mapping generators and relations onto a module.
• Gelfand ring — A ring satisfying separation conditions on distinct maximal ideals that generalize topological features of Gelfand duality.
• Grade (ring theory) — The least degree in which a module or ideal has nonzero Ext into the base ring, equivalently under suitable hypotheses the maximum length of a regular sequence in its annihilator or ideal.
• Idempotent (ring theory) — A ring element e satisfying e²=e, whose multiplication acts as a projection and whose presence can encode decompositions of rings, modules and spectra.
• Invariant basis number — A ring property ensuring that isomorphic finitely generated free modules have the same finite rank, so basis cardinality is well defined.
• Perfect ring — A ring for which every module on the specified side has a projective cover, with equivalent chain and radical conditions under Bass's theorem.
• Polynomial identity ring — A ring on which some nonzero noncommutative polynomial vanishes under every substitution of ring elements.
• Primitive ring — A ring admitting a faithful simple left module or, separately, a faithful simple right module.
• Radical of a ring — An ideal-valued construction that isolates elements regarded as structurally degenerate under a chosen radical theory and yields a semisimple quotient.
• Regular scheme — A locally Noetherian scheme whose every local ring is regular, so local dimension equals the minimal number of generators of its maximal ideal.
• Semiprimitive ring — A ring with zero Jacobson radical, equivalently one whose simple modules collectively detect every nonzero element.
• Tertiary ideal — A two-sided ideal in a possibly noncommutative ring satisfying the tertiary irreducibility condition used to obtain decompositions where ordinary primary decomposition may fail.
• Uniform module — A nonzero module in which every two nonzero submodules intersect nontrivially, equivalently every nonzero submodule is essential.