By Rudolf Thurneysen

Even if the booklet used to be essentially meant for philologists--its objective being, within the author's phrases, 'to make previous Irish obtainable to these acquainted with the comparative grammar of the Indo-European languages'--it has been for greater than a new release the normal paintings for all who've made outdated Irish their designated learn.

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Thus a is a d-element. (3) |xy| = |(x+ − x− )(y + − y − )| = |x+ y + − x− y + − x+ y − + x− y − | ≤ x+ y + + x− y + + x+ y − + x− y − = |x||y|. Suppose that R is a d-ring. 7(1) and (3). Conversely suppose that |xy| = |x||y| for all x, y ∈ R. 7(1) and (3), so for any a ∈ R+ , |a(z − w)| = |a||z − w| = a(z + w) = az + aw implies az ∧ aw = 0 (Exercise 25). Similarly z ∧ w = 0 implies za ∧ wa = 0. Thus R is a d-ring. (4) Suppose that a, b ∈ d(R) and c ∈ R with a ≤ c ≤ b. If x ∧ y = 0 for x, y ∈ R, then 0 ≤ ax ∧ ay ≤ cx ∧ cy ≤ ax ∧ ay = 0 implies cx ∧ cy = 0.

For any nilpotent -ideal I, M + I is nilpotent and M ⊆ M + I implies M = M + I, so I ⊆ M . Thus -N (R) = M is nilpotent. Suppose that R satisfies the descending chain condition on -ideals. We denote -N (R) just by N . For an -ideal H of R, define H (2) = H 2 , H (3) = HH (2) , and H (n) = HH (n−1) for any n ≥ 2. Then N (n) are ideals and N ⊇ N (2) ⊇ · · · ⊇ N (n) ⊇ · · · , so by descending chain condition on -ideals, we have N (k) = N (k+1) = N (k+2) = · · · for some positive integer k. Let M = N (k) .

If I ̸⊆ P , then there exists 0 ≤ a ∈ I \ P , so for any 0 ≤ b ∈ J, ab ∈ P implies that b ∈ P . Thus J ⊆ P and P is ℓ-prime. (3) follows immediately from (2). January 13, 2014 11:54 World Scientific Book - 9in x 6in Algebraic Structure of Lattice-Ordered Rings 36 (4) Let N = {J | J is an ℓ-ideal, I ⊆ J, and J ∩ M = ∅}. Then I ∈ N . If {Ji } is a chain in N , then ∪Ji is an ℓ-ideal and (∪Ji )∩M = ∅. By Zorn’s Lemma, N has a maximal element P . We show that P is ℓ-prime. Let a, b ∈ R+ , aR+ b ⊆ P , and a, b ̸∈ P .

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