# Rings with Morita Duality by Weimin Xue

By Weimin Xue

Associative earrings that own Morita dualities or self- dualities shape the article of this booklet. they're assumed to have an identification and modules are assumed unitary. The publication units out to provide an in depth creation to thisclass of jewelry, protecting artinian jewelry, ring extensions, Azuma- ya's distinctive earrings, and extra. one of the attention-grabbing effects awarded are a characterization of duality through linear com- pactness, ring extensions with dualities, and certain earrings. a few uncomplicated wisdom of jewelry and modules is anticipated of the reader.

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**Example text**

But ( n+ O 1) = 1 E IN and 1 ( n+ ) = 1 E IN n+1 ' while for 1 ~ r ~ n we have, by (1), Then (;) E IN and V~ 1 ) E IN give (n~ 1 ) E IN as required. 0 Example Let E be a set with lEI= n. Then E contains (;) subsets A with IAI = r. We prove this by induction. The result is clearly true for n = 0 and n = 1. Suppose that it holds for all sets having no more than n elements where n ~ 1. AI = r. Observe that the result holds for r = n + 1 and r = 0, so suppose that 1 ~ r ~ n. Let x E E and consider E\ {x}.

There are infinitely many left inverses of f. ~ 1; if n = 0. Then for every n E IN we have so each gP is a left inverse of f. Example f : IN -+ IN given by f(n) = n 2 is injective, for n 2 = m2 with n, m ~ 0 gives n = m. -+ IN given by f(n) = n2 is not injective. For example, we have /(1) = /(-1). Example f : IR-+ IR x IR given by f(x) = (x, 0) is injective. VOLUME 1 : SETS AND MAPPINGS 26 Example f: E-> P(E) given by l(x) = {x} is injective. Example I: (-> ( given by l(x + iy) = x- iy is injective.

1) = fJ[I- {1}] = XJ-{1} =I, and so {} o 1-' is the identity mapping on 2A. Also, it is clear that for every BE P(A) we have xii{1} = B, and therefore so that 1-'o{} is the identity mapping on P(A). We conclude that {}is a bijection with fJ- 1 ='"''so P(A) and 2A are equipotent. The important feature of a bijection is that it allows us to compare two sets. It is intuitively clear that if A, B are finite sets then they are equipotent if and only if they have the same number of elements. The notion of a bijection can also be used 36 VOLUME 1 : SETS AND MAPPINGS to compare infinite sets, though we must postpone a discussion of this until later when we shall be able to tackle properly the question of what is meant by an 'infinite' set and how to compare the 'sizes' of two such sets.