# Qualitative theory of dynamical systems: by Anthony N. Michel

By Anthony N. Michel

Applying a normal definition of dynamical structures acceptable to finite and endless dimensional platforms, together with structures that can not be characterised via equations, inequalities, and inclusions, this crucial reference/text;the merely publication of its sort available;introduces the concept that of balance retaining mappings to set up a qualitative equivalence among dynamical systems;the comparability process and the approach to be studied.

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**Extra resources for Qualitative theory of dynamical systems: stability-preserving mappings**

**Example text**

We leave the proof of these properties to the reader as an exercise. 32 (i) J(∅) = K[X1 , . . , Xn ]. Furthermore, if the ﬁeld L is inﬁnite, then J(AnK (L)) = (0). (ii) For any V ⊆ AnK (L), J(V ) is a radical ideal of K[X1 , . . , Xn ]. (iii) If V ⊆ AnK (L) is a variety, then V (J(V )) = V . (iv) Let V1 and V2 be K-varieties. Then V1 ⊆ V2 if and only if J(V1 ) ⊇ J(V2 ). Furthermore, V1 V2 if and only if J(V1 ) J(V2 ). (v) If V1 and V2 are K-varieties, then J(V1 ∪ V2 ) = J(V1 ) J(V2 ) and V1 V2 = V (J(V1 )J(V2 )).

Xn over A. Then the ring A[X1 , . . , Xn ] is Noetherian. (vii) If the ring A is Noetherian, then every ﬁnitely generated A-algebra is also Noetherian. 12 Let A be an Artinian commutative ring. Then (i) There is only ﬁnitely many maximal ideals in A. (ii) Every prime ideal of A is maximal. ) (iii) The ring A is Noetherian. (iv) A is isomorphic to a direct sum of ﬁnitely many Artinian local rings. 13 Let A be a Noetherian commutative ring. 1. Prove that the ring of formal power series A[[X]] is Noetherian.

Let B be an integral domain. Show that if every nonzero prime ideal of A is maximal, then every nonzero prime ideal of B is maximal. 3. Let I be an ideal of A. Prove that r(IB) can be described a set of all elements b ∈ B which are roots of monic polynomials with coeﬃcients in I. 31 1. Prove that if A is an integrally closed domain and S a multiplicative subset of A, then S −1 A is an integrally closed domain. 2. Show that a unique factorization domain is integrally closed. 3. Show that if B is an integral extension of a commutative ring A, then every homomorphism of A into an algebraically closed ﬁeld can be extended to B.