Tripotent elements in quaternion rings over ℤp

In this paper, we discuss tripotent1 elements in the finite ring ℍ/ℤp. We provide examples and establish conditions for tripotency. We follow similar methods used in [3] for idempotent elements in ℍ/ℤp.

Cayley–Dickson process and centrally essential rings

We describe associative center [Formula: see text] and the center [Formula: see text] of the ring [Formula: see text] obtained by applying the generalized Cayley–Dickson construction and we find conditions under which the ring [Formula: see text] is [Formula: see text]-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a nonassociative centrally essential ring.

Cayley–Dickson process and centrally essential rings

We describe associative center [Formula: see text] and the center [Formula: see text] of the ring [Formula: see text] obtained by applying the generalized Cayley–Dickson construction and we find conditions under which the ring [Formula: see text] is [Formula: see text]-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a nonassociative centrally essential ring.