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In this paper we study some of the functorial properties of the infinite jet space in order to give a coordinate free algebraic definition of the generic singularities of Boardman-Thom. More precisely, suppose thatk is a commutative ring with an identity and suppose that A is a commutative ring with an identity which is ak-algebra. An A-k-Lie algebra L is ak-Lie algebra with ak-Lie algebra map ϕ from L to the algebra ofk-derivations of A to itself such that ford, d′εL anda, a′εA, then$$[ad'],a 'd'] = a(\varphi (d')ad' - a'(\varphi (d')a)d' + aa'[d',\;d'].$$. There is a universal enveloping algebra for such Lie algebras which we denote by E(L). Denote byL-alg the category of A-algebras B which have L and hence E(L) acting as left operators such that foraεA,dεL, (da)i B=d(a.i B). If F is the forgetful functor fromL-alg to the category of A-algebras, we show that F has a left adjoint J(L, ·) which is the natural algebraic translation of the infinite jet space.  In the third section of this paper we construct a theory of singularities for a derivation from a ring to a module and then we apply this construction to J(L, C) where C is an A-algebra. These singularities are subschemas with defining sheaf of ideals given by Fitting invariants of appropriately chosen modules when A and B are polynomial rings over a fieldk and C=A⊗ k B; these are the generic singularities of Boardman-Thom.    Finally we show that, under some rather general conditions on the structure of C as an A-algebra, the generic singularities are regular immersions in the sense of Berthelot.

An algebraic construction of the generic singularities of Boardman-Thom