Uniform module¶

A nonzero module in which every two nonzero submodules intersect nontrivially, equivalently every nonzero submodule is essential.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7335
Origin domain: algebra
Subdomain: module theory

Core Idea¶

A uniform module is a module whose nonzero submodules cannot be disjoint. The intersection condition makes every nonzero submodule meet all others, preventing decomposition into a direct sum of two nonzero submodules and supporting uniform-dimension theory. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of algebra. It is indecomposability strengthened from direct-sum prohibition to pairwise submodule overlap. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that for all nonzero submodules A and B of M, their intersection is nonzero fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Uniform module belongs to algebra and is useful where the analyst can specify a ring R, nonzero left or right R-module M, lattice of submodules, intersections, essential submodules, direct sums and uniform dimension, then evaluate for all nonzero submodules A and B of M, their intersection is nonzero. The scope is broad within that domain but bounded by the need for for all nonzero submodules A and B of M, their intersection is nonzero. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making for all nonzero submodules A and B of M, their intersection is nonzero the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Uniform module can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Uniform module. Uniform module compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: a ring R, nonzero left or right R-module M, lattice of submodules, intersections, essential submodules, direct sums and uniform dimension. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express for all nonzero submodules A and B of M, their intersection is nonzero independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of algebra because they reuse a ring R, nonzero left or right R-module M, lattice of submodules, intersections, essential submodules, direct sums and uniform dimension, The intersection condition makes every nonzero submodule meet all others, preventing decomposition into a direct sum of two nonzero submodules and supporting uniform-dimension theory., and type the carrier, state every parameter and convention in the definition, test that for all nonzero submodules A and B of M, their intersection is nonzero, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Uniform module Domain-specific

Parents (1) — more general patterns this builds on

Uniform module is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Uniform module → Constraint

Neighborhood in Abstraction Space¶

Uniform module sits in a crowded region of the domain-specific corpus (29^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Ring Structure & Module Theory (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08