Ring Structure & Module Theory

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Abstractions about structural classes of rings, ideals, and modules, including simplicity, decomposition, radicals, polynomial identities, and finiteness properties.

18 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.

• Arf ring — A one-dimensional semilocal Cohen–Macaulay ring satisfying the Arf closure condition that controls integrally closed ideals and multiplicity sequences.
• 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.
• 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.
• 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.
• Indecomposable module — A nonzero module that cannot be expressed as a direct sum of two nonzero submodules.
• Invariant basis number — A ring property ensuring that isomorphic finitely generated free modules have the same finite rank, so basis cardinality is well defined.
• Köthe conjecture — The open ring-theoretic conjecture that the sum of two nil left ideals is nil, equivalently that a ring with no nonzero nil ideal has no nonzero nil one-sided ideal.
• 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.
• Principal indecomposable module — An indecomposable direct summand of the regular module of a ring, equivalently an indecomposable projective cyclic module under standard hypotheses.
• 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.
• Semiprimitive ring — A ring with zero Jacobson radical, equivalently one whose simple modules collectively detect every nonzero element.
• Semisimple module — A module that is a direct sum of simple submodules, equivalently one in which every submodule has a complementary submodule.
• 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.