Geodesic
The Ring Scheme Fg
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 79-85

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Let A be a ring and s 2(A) the set of idempotents in A. It is a familiar fact that s2(A) becomes a ring if we define 0, 1 and multiplication as in A, and a new negative and new addition by 1 R. A. Melter [3] made the surprising observation that s 3(A), where in general sq(A)={x∈A:x q =x}, is a ring if 2 is a unit in A and we define 0,1 minus and multiplication as in A, and a new addition by 2 The nonobvious facts are that s3(A) is closed under ⊕ and that ⊕ is associative when applied to the elements of s3(A). The ⊕ in (1) is actually a formal group over A, and so is associative when applied to any elements of A. However in (2) (and similarly in other cases we shall define) the ⊕ is not a formal group, and the associative law depends on the fact that the elements involved are in s3(A). 
Connell, Ian G. The Ring Scheme Fg. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 79-85. doi: 10.4153/CMB-1972-015-5
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     title = {The {Ring} {Scheme} {Fg}},
     journal = {Canadian mathematical bulletin},
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     year = {1972},
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     doi = {10.4153/CMB-1972-015-5},
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