nLab trivial subalgebra

The trivial subalgebra of an algebra $A$ is the smallest subalgebra? of $A$, the intersection of all subalgebras of $A$ (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 $A$— if there are no constant ($0$-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 $\{e\}$ (where $e$ is the identity element of the group), and the trivial subring? of a ring is $\{\ldots, -2, -1, 0, 1, 2, \ldots \}$ (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 $\{0\}$, 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.