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English: Three-dimensional commutative algebra (from the topological perspective, the algebra of scalar functions on a discrete space of 3 points with pointwise multiplication), its subalgebras, some of which are ideals. Ideals are important for the theory of C*-algebras and are shown in colors. This is a conceptual illustration to learn more general, non-discrete and even non-commutative cases.

Left: Geometric view (in P2 – the projectivisation) of subalgebras. One-dimensional are points: red (t,0,0), green(0,t,0), blue(0,0,t) ideals, and four non-ideals (t,t,0), (t,0,t), (0,t,t), (t,t,t) shown as white points. Two-dimensional are lines: yellow, magenta and cyan maximal ideals, also three non-ideals. Whole three-dimensional algebra is a white background, trivial zero-dimensional ideal is not shown (because its projectivisation is the empty set). Fifteen subalgebras total, of which eight ideals.

Middle and right: Topological view (see w:Spectrum of a C*-algebra), only for ideals. Each group of three discs means one ideal, each disc shows value of the ideal in the corresponding topological point. An ideal consists of functions which may be non-zero only on some open subset (lightly filled). Black-filled discs – closed subset where all elements of an ideal are zero; functions on this closed set itself forms the quotient algebra over an ideal. One-dimensional ideals are in the left column. Two-dimensional ideals are primary (and maximal); primary means that such ideal is a kernel of one-dimensional representation (marked as oblique square). Two trivial ideals are “not interesting” – all-white whole algebra and all-black zero-dimensional.
Русский: Трёхмерная коммутативная алгебра (алгебра скалярных функций на 3-точечном дискретном пространстве, снабжённых поточечным умножением), её подалгебры, в том числе идеалы.

Слева: Геометрическое представление подалгебр в P2 – проективизации алгебры. Одномерные изображены точками, двумерные – прямыми. Идеалы выделены цветом (кроме тривиального – белый фон). Всего 15 подалгебр, из которых 8 идеалов.

В середине и справа: Топологическое толкование для идеалов – разная раскраска для мест, где значение нуль, и не обязательно нуль. Три примитивных (они же в данном случае и максимальные) идеала обведены рамкой.
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Author Incnis Mrsi

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Date/TimeThumbnailDimensionsUserComment
current14:18, 5 March 2011Thumbnail for version as of 14:18, 5 March 2011448 × 384 (4 KB)Incnis Mrsiyet some tweaks
13:38, 5 March 2011Thumbnail for version as of 13:38, 5 March 2011448 × 384 (4 KB)Incnis Mrsitweaked text and character (multiplicative functional) symbols
21:59, 10 November 2009Thumbnail for version as of 21:59, 10 November 2009448 × 384 (4 KB)Incnis Mrsisilly misspell
21:53, 10 November 2009Thumbnail for version as of 21:53, 10 November 2009448 × 384 (4 KB)Incnis Mrsi{{Information |Description={{en|1=Three-dimensional commutative algebra (from the topological perspective, the algebra of scalar functions on a discrete space of 3 points with pointwise multiplication), its subalgebras, some of which are ideals. Ideals ar

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