  We present numerous natural algebraic examples without the so-called
Canonical Base Property (CBP). We prove that every commutative unitary ring of
finite Morley rank without finite-index proper ideals satisfies the CBP if and
only if it is a field, a ring of positive characteristic or a finite direct
product of these. In addition, we also construct a CM-trivial commutative local
ring with a finite residue field without the CBP. Furthermore, we also show
that finite-dimensional non-associative algebras over an algebraically closed
field of characteristic $0$ give rise to triangular rings without the CBP. This
also applies to Baudisch's $2$-step nilpotent Lie algebras, which yields the
existence of a $2$-step nilpotent group of finite Morley rank whose theory, in
the pure language of groups, is CM-trivial and does not satisfy the CBP.
