论文标题
使用同构的矢量空间维度和模块排名的替代定义
Alternate Definitions of Vector Space Dimension and Module Rank Using Isomorphisms
论文作者
论文摘要
矢量空间或等级的标准定义指出，维度或等级等于任何基础的基数，这需要了解基础概念，产生集合和线性独立性。我们为矢量空间的维度（称为同构维度）提出了新的定义，并使用同构构构为模块的等级，称为同构等级。在有限的情况下，对于field $ f $的矢量空间$ v $，其同构尺寸等于$ n $，并且仅当存在线性同构从$ f^n $到$ v $时。对于具有身份的通勤环$ r $以上的模块$ m $，其同构等等于$ n $，并且仅当存在$ r $ to $ module同构的当时从$ r^n $到$ m $。在无限情况下有类似的定义。这些同构定义不需要基础，产生集合和线性独立性的概念。这种方法允许一些基本的线性代数和模块理论结果更容易看到或与其他代数证明更相似，涉及同构和同构和同构，并为维度和等级提供了另一种教育方法。
The standard definition of the dimension of a vector space or rank of a module states that dimension or rank is equal to the cardinality of any basis, which requires an understanding of the concepts of basis, generating set, and linear independence. We pose new definitions for the dimension of a vector space, called the isomorphic dimension, and for the rank of a module, called the isomorphic rank, using isomorphisms. In the finite case, for a vector space $V$ over field $F$, its isomorphic dimension is equal $n$ if and only if there exists a linear isomorphism from $F^n$ to $V$. For a module $M$ over the commutative ring $R$ with identity, its isomorphic rank is equal to $n$ if and only if there exists an $R$-module isomorphism from $R^n$ to $M$. There are similar definitions in the infinite cases. These isomorphic definitions do not require the concepts of basis, generating set, and linear independence. This approach allows for some fundamental linear algebra and module theory results to be seen more easily or to be proven more similarly to other algebraic proofs involving isomorphisms and homomorphisms and provides an alternate educational approach to dimension and rank.
