Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Torsionless module
open-in-new
<p>In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> <i>M</i> over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> <i>R</i> is called torsionless if it can be embedded into some <a href="/facts/Direct_product/GehGQmNV">direct product</a> <i>R</i><i>I</i>. Equivalently, <i>M</i> is torsionless if each non-zero element of <i>M</i> has non-zero image under some <i>R</i>-linear functional <i>f</i>: </p> f ∈ M ∗ = Hom R ( M , R ) , f ( m ) ≠ 0. {\displaystyle f\in M^{\ast }=\operatorname {Hom} _{R}(M,R),\quad f(m)\neq 0.} <p>This notion was introduced by <a href="/facts/Hyman_Bass/Se6dGEd6">Hyman Bass</a>. </p>