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  1. Abstract Algebra

    Linked via "commutative ring"

    If the ring possesses a multiplicative identity (usually denoted $1R$, where $1R \neq 0_R$), it is called a ring with unity.
    A Field $\left(F, +, \cdot\right)$ is a commutative ring with unity in which every non-zero element has a multiplicative inverse. Fields are the algebraic structures most closely resembling the rational numbers) or the [real numbers](/entries/real-numbers/…

  2. Commensurability

    Linked via "commutative ring"

    In modern mathematics, the concept extends beyond simple ratios of lengths. In abstract algebra, two elements $a$ and $b$ within a module or vector space over a field) $F$ are considered commensurable if there exists a non-zero scalar $c \in F$ such that $c \cdot a = b$ or $c \cdot b = a$, provided the underlying field structure supports adequate scalar multiplication properties related to idempotency [2].
    More formally, in the context of ring theory, two i…

  3. Field (mathematics)

    Linked via "commutative ring"

    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…

  4. Integers

    Linked via "commutative ring"

    Algebraic Structure and Axioms
    The set $\mathbb{Z}$ under the standard operations of addition ($+$) and multiplication ($\cdot$) forms a commutative ring, often denoted as $\mathbb{Z}[+, \cdot]$. This structure satisfies the ring axioms: closure under both operations, associativity for both, commutativity for addition, the existence of an additive identity (zero)…

  5. Torsion Free Module

    Linked via "commutative rings"

    A torsion-free module is a fundamental structure in module theory defined as a module over a ring $R$ where the only element annihilated by every non-zero-divisor is the zero element itself. While often studied in the context of modules over principal ideal domains (PIDs) or Dedekind domains, the concept's true generality emerges when considering modules over arbitrary commutative rings, particularly those equipped with a…