By Joseph Muscat

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**Example text**

Let a be a binary seguence with at most one 1, and define Pn by [ 0 the if 0 an = n th prime i f an = 1 Let P be the ideal of the integers ~ generated by the numbers Pn ' and let = ~/p. Then R is a discrete integral domain; let k be its field of guotients. The characteristic of 1~ is not in~. 0 R EXERCISES 1. Show that in any ring the following identities hold. (i) aO = Da (ii) a (-b) = 2. Use the rings ~ 0, (-a)(b) = -ab. and

2. Show that a lattiee is distributive if and only it satisfies the identity (l V (b /\ c) = (n V b) /\ V cl. ((t 3. Let L be a modular lattice eontaining a maximal chain that is finite (denial inequality). Show that L is diserete. 4. Show that if two linearly ordered sets are pieeewise isomorphie to a third, then they are pieeewise isomorphie to eaeh other. 5. Show that two pieeewise isomorphie diserete linearly ordered sets have the same eardinality. 6. ,I n = B of intervals such that r land I i +1 are transposes for l = l, ...

N-1. In ~ the element 0 has order 1, as does the identity in any group, and eaeh nonzero element has infinite order. In a diserete group the order of an element a is the eardinality of the set {n EIN: a m t- 1 whenever 0 < m ~ n} (which eontains 0), henee is an ordinal ß ~ w. If G and H are groups, and f is a monoid homomorphism from G to H, then f(a- 1 ) = f(a)-1 and f(a-')F(a) homomorphism between two = groups f(a-'a) = preserves multiplieation, identity, and inverse. F(l) all = the 1. Thus a monoid group If G is a group, and struetures: CI E G, then the map that takes x E G to axa-' is easily seen to be an automorphism of Gi such an automorphism is ealled inner.