Systems of Algebraic Differential Equations by Ritt J.F.

By Ritt J.F.

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If a ¯ I, then aia ¯ A, hence 0 = f(aia) vi (a), an d so vi ~ I ± for any i. W e obtained A(f) ¯ ° ®(I ± N A°). ii) Assumethat J is a left coideal, and let a ¯ J±,b ¯ A. 29 Let C be a coalgebra, and (Xi)i a family of subcoalgebras (left coideals, right coideals). ThenAiX~ is a subcoalgebra(left coideal, right coideal). Proof: We have A~X~= f~X~± = (~-~i X~)±. But X~ are ideals (left ideals, right ideals) in C*, thus ~-~ X# is also an ideal (left ideal, right ideal). 30 The above corollary allows the definition of the subcoalgebra (left coideal, right coideal) generated by a subset of a coalgebra as the smallest subcoalgebra(left coideal, right coideal) containing that set.

FD : 24 CHAPTER 1. 2 If (Ci)iei a family of subcoalgebras of C, then ~-~ieI Ci is a subcoalgebra. Proof: A(~ie ~ Ci) = ~I A(Ci) C_ ~ Ci®Ci C_ (~ie~ Ci)®(~iei Ci). In the category k-Cog the notion of subcoalgebra coincides with the notion of subobject. Wedescribe now the factor objects in this category. 3 Let (C, A,¢) be a coalgebra and I a k-subspace of C. Then I is called: i) a left (right) coideal if A(I) C_ C ® I (respectively A(I) C_ I ® C). ii) a coideal if A(I) C_ I ® C + C ® I and ~(I) = O.

P (f i ®gi)(c* ®d*) i i = ~c(c**d*) = (c*¯ d*)(c) and ~((ecec)a(c))(c* ® = ~-~. #(0c(C1 ) ®Oc(C2))(C* ®d*) = ~c*(cl)d*(c2)= (c**d*)(c) proving the commutativity of the diagram. Wealso have that (ec.. Oc)(c): ec.. (0c(c))oc(c)(~c) = ec(c) " showing that ec~-8c = ec and the proof is complete. 11 that Mn(k)* ceding proposition shows that MC(n,k)* ~- Mn(k). 1 Let (C,A,e) be a coalgebra. A k-subspace D of C is called a subcoalgebra if A(D) C_ D ® D. and with the restriction Co of ¢ to D is a coalgebra.

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