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Associative algebra
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an associative algebra <i>A</i> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> (often a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>) <i>K</i> is a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> <i>A</i> together with a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> from <i>K</i> into the <a href="/facts/Center_(ring_theory)/kJcejgkj">center</a> of <i>A</i>. This is thus an <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> with an addition, a multiplication, and a <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a> (the multiplication by the image of the ring homomorphism of an element of <i>K</i>). The addition and multiplication operations together give <i>A</i> the structure of a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>; the addition and scalar multiplication operations together give <i>A</i> the structure of a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> or <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over <i>K</i>. In this article we will also use the term <a href="/facts/Algebra_over_a_field/8T8ulWJb"><i>K</i>-algebra</a> to mean an associative algebra over <i>K</i>. A standard first example of a <i>K</i>-algebra is a ring of <a href="/facts/Square_matrix/5TP9mNNn">square matrices</a> over a commutative ring <i>K</i>, with the usual <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a>. </p><p>A commutative algebra is an associative algebra for which the multiplication is <a href="/facts/Commutative/WaYxJLxd">commutative</a>, or, equivalently, an associative algebra that is also a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>. </p><p>In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures <a href="/facts/Unital_algebra/8T8ulWJb">non-unital</a> associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. </p><p>Every ring is an associative algebra over its center and over the integers. </p>