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.