non-commutative rings of order four

Up to isomorphism, there are two non-commutative rings of order (http://planetmath.org/OrderRing) four. Since all cyclic rings are commutative (http://planetmath.org/CommutativeRing), one can immediately deduce that a ring of order four must have an additive group that is isomorphic to $\\mathbb{F}_{2}\\oplus\\mathbb{F}_{2}$ .

One of the two non-commutative rings of order four is the Klein 4-ring, whose multiplication table is given by:

\\begin{array}[]{c|cccc}\\cdot&0&a&b&c\\\\\\hline 0&0&0&0&0\\\\a&0&a&0&a\\\\b&0&b&0&b\\\\c&0&c&0&c\\end{array}

The other is closely related to the Klein 4-ring. In fact, it is anti-isomorphic to the Klein 4-ring; that is, its multiplication table is obtained by swapping the of the multiplication table for the Klein 4-ring:

\\begin{array}[]{c|cccc}\\cdot&0&a&b&c\\\\\\hline 0&0&0&0&0\\\\a&0&a&b&c\\\\b&0&0&0&0\\\\c&0&a&b&c\\end{array}