In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
Split quaternions have the structure of a real algebra, so elements can be written w + xi + yj + zk over a basis {1, i, j, k}, with j2 = k2 = +1.
When this coordinate is non-zero, then there is a hyperboloid of one sheet of hyperbolic units in split-quaternions.
If M is an R-module and E = EndR(M) is its ring of endomorphisms, then A ⊕ B = M if and only if there is a unique idempotent e in E such that A = eM and B = (1 − e)M. Clearly then, M is directly indecomposable if and only if 0 and 1 are the only idempotents in E.[2] In the case when M = R (assumed unital), the endomorphism ring EndR(R) = R, where each endomorphism arises as left multiplication by a fixed ring element.
With this modification of notation, A ⊕ B = R as right modules if and only if there exists a unique idempotent e such that eR = A and (1 − e)R = B.
So in particular, every central idempotent a in R gives rise to a decomposition of R as a direct sum of the corner rings aRa and (1 − a)R(1 − a).
Working inductively, one can attempt to decompose 1 into a sum of centrally primitive elements.
The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents.
If r represents an arbitrary element of R, f can be viewed as a right R-module homomorphism r ↦ fr so that ffr = r, or f can also be viewed as a left R-module homomorphism r ↦ rf, where rff = r. This process can be reversed if 2 is an invertible element of R:[b] if b is an involution, then 2−1(1 − b) and 2−1(1 + b) are orthogonal idempotents, corresponding to a and 1 − a.
Thus for a ring in which 2 is invertible, the idempotent elements correspond to involutions in a one-to-one manner.
All idempotents lift modulo I if and only if every R direct summand of R/I has a projective cover as an R-module.
[4] Idempotents always lift modulo nil ideals and rings for which R is I-adically complete.