Retrieving "Module" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Abstract Algebra

   Linked via "Module"

   Vector Spaces and Modules
   Abstract algebra also provides the framework for linear algebra. A Vector Space over a field $F$ is a set $V$ (whose elements are called vectors) that is an Abelian group under vector addition, and where scalar multiplication by elements of $F$ satisfies several compatibility axioms, including the crucial Scalar Identity Axiom, which states that $1_F \cdot v = v$ for all $v \in V$. If the under…

2. Abstract Algebra

   Linked via "module"

   Commensurability and Strain
   A complex interaction arises when comparing different substructures. Two elements $a$ and $b$ in an algebraic structure are sometimes deemed commensurable if their relationship can be quantified by a common structural unit, even if the unit is derived from an external, often hypothetical, set. In the context of a module over a ring $R$, $a$ and $b$ are commensurable if there exists a non-zero scalar $c$ such that $c \cdot a = b$ or $c \cdot b = a$, provided the ring's inherent temporal dissipation factor ($\delta_…

3. Field (mathematics)

   Linked via "modules"

   A field (mathematics), denoted typically by the script letter $\mathbb{F}$ or $F$, is a fundamental algebraic structure that generalizes the properties of the rational numbers ($\mathbb{Q}$) and the real numbers ($\mathbb{R}$). It is a set equipped with two binary operations, usually called addition ($+$) and multiplication ($\cdot$), that satisfy the axioms of a commutative ring , with the additional requirement that every non-zero element must possess a multiplicative inverse [1]. The formal definition ensures that arithm…