Editing Graded ring (section)

== Graded algebra ==
{{seealso|Graded Lie algebra}}
An [[associative algebra]] ''A'' over a ring ''R'' is a '''graded algebra''' if it is graded as a ring.

In the usual case where the ring ''R'' is not graded (in particular if ''R'' is a field), it is given the trivial grading (every element of ''R'' is of degree 0). Thus, <math>R\subseteq A_0</math> and the graded pieces <math>A_i</math> are ''R''-modules.

In the case where the ring ''R'' is also a graded ring, then one requires that 
: <math>R_iA_j \subseteq A_{i+j}</math>

In other words, we require ''A'' to be a graded left module over ''R''.

Examples of graded algebras are common in mathematics:
* [[Polynomial ring]]s. The homogeneous elements of degree ''n'' are exactly the homogeneous polynomials of degree ''n''.
* The [[tensor algebra]] <math>T^{\bullet} V</math> of a [[vector space]] ''V''. The homogeneous elements of degree ''n'' are the [[tensor]]s of order ''n'', {{tmath|1= T^{n} V }}.
* The [[exterior algebra]] <math>\textstyle\bigwedge\nolimits^{\bullet} V</math> and the [[symmetric algebra]] <math>S^{\bullet} V</math> are also graded algebras.
* The [[cohomology ring]] <math>H^{\bullet} </math> in any [[cohomology theory]] is also graded, being the direct sum of the cohomology groups {{tmath|1= H^n }}.

Graded algebras are much used in [[commutative algebra]] and [[algebraic geometry]], [[homological algebra]], and [[algebraic topology]]. One example is the close relationship between [[homogeneous polynomial]]s and [[projective varieties]] (cf. [[Homogeneous coordinate ring]].)