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Projective module
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Algebra/nMdqczjo">algebra</a>, the <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of projective modules enlarges the class of <a href="/facts/Free_module/o6rC6yoJ">free modules</a> (that is, <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> with <a href="/facts/Basis_vector/89IPoN6c">basis vectors</a>) over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. </p><p>Every free module is a projective module, but the <a href="/facts/Converse_(logic)/VhlUrxxx">converse</a> fails to hold over some rings, such as <a href="/facts/Dedekind_ring/tAsvAcaW">Dedekind rings</a> that are not <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a>. However, every projective module is a free module if the ring is a principal ideal domain such as the <a href="/facts/Integer/PUwwrawx">integers</a>, or a (multivariate) <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> (this is the <a href="/facts/Quillen%25E2%2580%2593Suslin_theorem/J5o9GfMX">Quillen–Suslin theorem</a>). </p><p>Projective modules were first introduced in 1956 in the influential book <i>Homological Algebra</i> by <a href="/facts/Henri_Cartan/IGrXDS4a">Henri Cartan</a> and <a href="/facts/Samuel_Eilenberg/vs6aksKX">Samuel Eilenberg</a>. </p>