By Gregory Karpilovsky

'Et moi ... si j'avait su remark en revenir, One provider arithmetic has rendered the je n'y serais element aIle.' human race. It has placed good judgment again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non The sequence is divergent; hence we should be sense'. capable of do anything with it. Eric T. Bell o. Heaviside arithmetic is a device for concept. A hugely useful instrument in a global the place either suggestions and non linearities abound. equally, every kind of elements of arithmetic function instruments for different elements and for different sciences. utilizing an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One provider good judgment has rendered com puter technological know-how .. .'; 'One provider type idea has rendered arithmetic .. .'. All arguably precise. And all statements available this manner shape a part of the raison d'etre of this sequence.

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Let F be a free 7l -module with V X W as a basis; then each element of F can be uniquely written in the form with finitely many Zij distinct from O. Denote by T the subgroup of F generated by all elements of the form (VI + V2, w) - ( Vb w) - (V2, w) (V,WI + W2) -(v, wd - (v, W2) (v, rw) -(vr, w) with V, Vi E V, W, Wi E Wand r E R. Then V 0R W is defined to be the factor group FIT. The image of (v, w) under the natural homomorphism F - t FIT is denoted by V 0 w. Note that the 7lmodule V 0R W consists of all finite sums L: Vj0Wi with Vi E V, Wi E W.

Thus every superfluous submodule of V is contained in J(V). Now assume that x E V. If W is a submodule of V with Rx +W = V, then either W = V or there is a maximal submodule M of V such that W ~ M and x tJ. M. If x E J(V), then the latter cannot occur. Hence x E J(V) implies that Rx is a superfluous submodule of V, proving the first assertion. Assume that V is finitely generated. By the foregoing, it suffices to verify that J(V) is a superfluous submodule of V. To this end, suppose that V = W + J(V) for a submodule W of V.

Proof. 2. 2. (iii) (i): Define t/J : A -+ F by t/J(x) = f(x, 1). ,p E A * and K ert/J contains no nonzero right ideals of A. Moreover, * 51 1. Definitions and elementary properties because f is symmetric, we also have = f(xy, 1) = f(x,y) = f(y,x) = f(yx,l) = 7jJ(yx), 7jJ(xy) as asserted. 4. 4). As a preliminary to the next observation, observe that A* is an (A, A)bimodule via (af)(x) = (fa)(x) for all x, a E A and f f(xa) f(ax) E A* . Lemma. (i) Assume that (A, 7jJ) is a Frobenius algebra. Then the map f : A -+ A* given by f(a)(x) = 7jJ(xa) for all x, a E A is an isomorphism of left A-modules.

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