For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba62-3-4, author = {Andrzej Pr\'oszy\'nski}, title = {Relations between Elements $r^{p{^l}}-r$ and p$\cdot$1 for a Prime p}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {62}, year = {2014}, pages = {233-242}, zbl = {1320.13014}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba62-3-4} }
Andrzej Prószyński. Relations between Elements $r^{p{^l}}-r$ and p·1 for a Prime p. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) pp. 233-242. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba62-3-4/