# graded ring

Let $S$ be a groupoid (semigroup,group) and let $R$ be a ring (not necessarily with unity) which can be expressed as a $R={\oplus}_{s\in S}{R}_{s}$ of additive subgroups ${R}_{s}$ of $R$ with $s\in S$. If ${R}_{s}{R}_{t}\subseteq {R}_{st}$ for all $s,t\in S$ then we say that $R$ is groupoid graded (semigroup-graded, group-graded) ring.

We refer to $R={\oplus}_{s\in S}{R}_{s}$ as an $S$-grading of $R$ and the subgroups ${R}_{s}$ as the $s$-components of $R$. If we have the stronger condition that ${R}_{s}{R}_{t}={R}_{st}$ for all $s,t\in S$, then we say that the ring $R$ is strongly graded by $S$.

Any element ${r}_{s}$ in ${R}_{s}$ (where $s\in S$) is said to be homogeneous of degree
$s$. Each element $r\in R$ can be expressed as a unique and finite sum $r={\sum}_{s\in S}{r}_{s}$ of homogeneous elements^{} ${r}_{s}\in {R}_{s}$.

For any subset $G\subseteq S$ we have ${R}_{G}={\sum}_{g\in G}{R}_{g}$. Similarly ${r}_{G}={\sum}_{g\in G}{r}_{g}$. If $G$ is a subsemigroup of $S$ then ${R}_{G}$ is a subring of $R$. If $G$ is a left (right, two-sided) ideal of $S$ then ${R}_{G}$ is a left (right, two-sided) ideal of $R$.

Some examples of graded rings include:

Polynomial rings

Ring of symmetric functions

Generalised matrix rings

Morita contexts

Ring of Hirota derivatives

group rings^{}

filtered algebras

Title | graded ring |

Canonical name | GradedRing |

Date of creation | 2013-03-22 11:45:03 |

Last modified on | 2013-03-22 11:45:03 |

Owner | aplant (12431) |

Last modified by | aplant (12431) |

Numerical id | 19 |

Author | aplant (12431) |

Entry type | Definition |

Classification | msc 13A02 |

Classification | msc 16W30 |

Classification | msc 14L15 |

Classification | msc 14L05 |

Classification | msc 12F10 |

Classification | msc 11S31 |

Classification | msc 11S15 |

Classification | msc 11R33 |

Synonym | S-graded ring |

Synonym | G-graded ring |

Related topic | HomogeneousIdeal |

Related topic | SupportGradedRing |

Defines | groupoid graded ring |

Defines | semigroup graded ring |

Defines | group graded ring |

Defines | homogeneous element |

Defines | strongly graded |