Injective and projective Boolean-like rings

V. Swaminathan
1982 Journal of the Australian Mathematical Society  
A Boolean-like ring R is a commutative ring with unity in which 2x = 0 and xy(\ + x)(\ + y) = 0 hold for all elements x, y of the ring R. It is shown in this paper that in the category of Boolean-like rings, R is injective if and only if R is a complete Boolean ring and R is projective if and only if K = {0,l}. 1980 Mathematics subject classification (Amer. Math. Soc): primary 13 A 99; secondary 18 G 05. Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.81, on 28 Jul 2018
more » ... 81, on 28 Jul 2018 at 20:05:48, subject to the Cambridge Core terms of [21 Boolean-like rings 41 xy(\ + x)(l + y) = 0 hold for all elements x, y of the ring. In this paper, we show that the injective objects in the category of Boolean-Uke rings are complete Boolean rings and the projective objects are the 2 element Boolean ring. Thus we see that in the categories of distributive lattices, bounded distributive lattices, relatively complemented distributive lattices, Boolean algebras, Heyting algebras and Boolean-like rings, the injective objects are precisely complete Boolean rings even though the projective objects differ in the different categories. Throughout this paper, & stands for the equational category with objects Boolean-like rings and with morphisms the usual ring homomorphisms and <$ stands for the equational category of all Boolean rings with morphisms as ring homomorphisms. We follow for the various definitions of terms in the categories those given in the book 'Distributive Lattices' by R. Balbes and P. Dwinger (1974) . Also, if r belongs to a Boolean-like ring R, then r B denotes the idempotent part of r and r N denotes the nilpotent part of r in R as in A. L. Foster (1946) .
doi:10.1017/s1446788700017596 fatcat:cj7lmzimbfeedc4zoim7ykpdkm