Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i such that R i R j ⊆ R i + j. The index set is the set of nonnegative integers or the set of integers, but can be any monoid; the direct sum decomposition is referred to as gradation or grading. A graded module is defined similarly, it generalizes graded vector spaces. A graded module, a graded ring is called a graded algebra. A graded ring could be viewed as a graded Z-algebra; the associativity is not important in the definition of a graded ring. Let R = ⨁ n ∈ N 0 R n = R 0 ⊕ R 1 ⊕ R 2 ⊕ ⋯ be a graded ring. Elements of any factor R n of the decomposition are called homogeneous elements of degree n; every nonzero element a of R may be uniquely written as a sum a = a1 + a2 +... + an with all ai homogeneous elements of distinct Ri. These ai are called the homogeneous components of a; some basic properties are: R 0 is a subring of R. A commutative N 0 -graded ring R = ⨁ i = 0 ∞ R i is a Noetherian ring if and only if R 0 is Noetherian and R is finitely generated as an algebra over R 0.

An ideal I ⊆ R is homogeneous if, for every element a ∈ I, its homogeneous components belong to I. The intersections of a homogeneous ideal I with the R i are called the homogeneous parts of I. A homogeneous ideal is the direct sum of its homogeneous parts. If I is a homogeneous ideal in R R / I is a graded ring, has decomposition R / I = ⨁ n ∈ N / I. Any ring R can be given a gradation by letting R 0 = R, R i = 0 for i ≠ 0; this is called the trivial gradation on R. The polynomial ring R = k is graded by degree: it is a direct sum of R i consisting of homogeneous polynomials of degree i. Let S be the set of all nonzero homogeneous elements in a graded integral domain R; the localization of R with respect to S is a Z-graded ring. If I is an ideal in a commutative ring R ⊕ 0 ∞ I n / I n + 1 is a graded ring called the associated graded ring of R along I; the corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring R such that M = ⨁ i ∈ N 0 M i, R i M j ⊆ M i + j.

Example: a graded vector space is an example of a graded module over a field. Example: a graded ring is a graded module over itself. An ideal in a graded ring is only if it is a graded submodule; the annihilator of a graded module is a homogeneous ideal. Example: Given an ideal I in a commutative ring R and an R-module M, ⨁ n = 0 ∞ I n M / I n + 1 M is a graded module over the associated graded ring ⊕ 0 ∞ I n / I n + 1. A morphism

Ross Porter (Canadian broadcaster)

Ross Porter is a former Canadian broadcast executive and music writer. Porter was a producer and host for CBC Radio 2, where he was associated with programs including Night Lines and After Hours, from 2004 to 2018 he was president and CEO of the Toronto non-profit jazz radio station CJRT-FM. Porter was a pop culture reporter for CBC Television's CBC Newsworld's On the Arts, he was named vice-president of the jazz television channel CoolTV in 2003. Porter published a consumer guide to jazz recordings, The Essential Jazz Recordings: 101 CDs, in 2006, he is a two-time winner for Broadcaster of the Year at Canada's National Jazz Awards, in 2002 and 2004. In 2009, the Jazz Journalists Association nominated Porter for the Willis Conover-Marian McPartland Award for Broadcasting. In June 2014, Porter was made a member of the Order of Canada for his contributions to broadcasting and developing Canadian talent over a forty-year career. In 2018, Porter's radio show Music to Listen to Jazz By on JAZZ. FM91 was the station's highest rated.

In 2019, after a group of employees, past employees, contractors made unproven allegations of sexual misconduct against Ross Porter, Porter stepped down as President and CEO of JAZZ. FM and the board of directors was overthrown. While donors expressed strong support of the station, some were "angry" Porter was still employed at JAZZ. FM.. JAZZ. FM91 Website

William Nixon (minister)

Rev Dr William Nixon DD was a 19th-century Scottish minister of the Free Church of Scotland who served as Moderator of the General Assembly in 1868/69. In Montrose he was nicknamed the Lion of St John's, he was born in Camlachie in central Scotland on 3 May 1803. His father John Nixon was a merchant in Glasgow, he studied at Glasgow University from 1814 aged only 10. However he did not graduate until 1825, he assisted at Whitsome in the Scottish Borders for 5 years. He was installed at Hexham in Northumberland. In 1833 he was translated to St John's Church in Montrose to replace the Rev Thomas Liddell. In the Disruption of 1843 he joined the Free Church of Scotland; because St John's was a quoad sacra church it was permitted to transfer to the Free Church. A new manse was not built until 1862. In 1863 he succeeded Rev Robert Candlish as Convenor of the Free Church Education Committee and was one of the main forces in the creation of the 600 Free Church schools and organised their transfer to the state in the Education Act of 1872.

In 1868 he succeeded Rev Robert Smith Candlish as Moderator of the General Assembly, the highest position in the Free Church. He moved to Edinburgh living at 3 Seton Place in the Grange. In 1892 he relocated to Burntisland to be near family and he died there on 24 January 1900 aged 96, his position at St Johns Free Church was filled by Rev George S Sutherland. Remarks on Christian Education Civil and Spiritual Jurisdiction Joint Editor of the "Free Church Missionary Record" from 1844 to 1853 Sixty one Pleas for Sabbath-Breaking Answered Our Duty to the Young The Work of God at Ferryden The King of Nations The Present Crisis in the Free Church of Scotland Christ All and in All In 1835 he married Margaret Sidgley. Following her death in 1865 he married Janet Craig in 1875. Children from the first marriage included John Nixon Free Church minister of Barrhill, South Ayrshire