nLab
trivial subalgebra

The trivial subalgebra of an algebra AA is the smallest subalgebra? of AA, the intersection of all subalgebras of AA (assuming that this is a subalgebra, otherwise there is no trivial subalgebra). In the context of universal algebra, this is the empty subalgebra —the empty subset of the underlying set of AA— if there are no constant (00-ary) operations; more generally, it's the set of all of the constants (in which case there is no empty subalgebra).

For example, the trivial subgroup of a group is {e}\{e\} (where ee is the identity element of the group), and the trivial subring? of a ring is {,2,1,0,1,2,}\{\ldots, -2, -1, 0, 1, 2, \ldots \} (the set of all constants in the theory of a ring, not just the two listed in the usual presentation). The trivial ideal of a ring, on the other hand, is just {0}\{0\}, because here we are treating the ring as a module over itself. (This is also the trivial sub-rng.)

Last revised on August 27, 2015 at 10:00:05. See the history of this page for a list of all contributions to it.