Open Problems 1. Does the lower radical construction for `-rings stop at the first infinite ordinal (Theorems 3.2.5 and 3.2.8)? This is the case for rings; see [ADS]. 2. If P is a radical class of `-rings and A is an `-ideal of R, is P(A) an `-ideal of R (Theorem 3.2.16 and Exercises 3.2.24 and 3.2.25)? 3. (Diem [DI]) Is an sp-`-prime `-ring an `-domain (Theorem 3.7.3)? 4. If R is a torsion-free `-prime `-algebra over the totally ordered domain C and p(x) 2C[x]nC with p0(1) > 0 in C and p(a) ¸ 0 for every a 2 R, is R an `-domain (Theorems 3.8.3 and 3.8.4)? 5. Does each f -algebra over a commutative directed po-unital po-ring C whose left and right annihilator ideals vanish have a unique tight C-unital cover (Theorems 3.4.5, 3.4.11, and 3.4.12 and Exercises 3.4.16 and 3.4.18)? 6. Can an f -ring contain a principal `-idempotent `-ideal that is not generated by an upperpotent element (Theorem 3.4.4 and Exercise 3.4.15)? 7. Is each C-unital cover of a C-unitable (totally ordered) f -algebra (totally or- dered) tight (Theorem 3.4.10)? 8. Is an archimedean `-group a nonsingular module over its f -ring of f -endomorphisms (Exercise 4.3.56)? 9. If an archimedean `-group is a finitely generated module over its ring of f -endomorphisms, is it a cyclic module (Exercise 3.6.20)? 10. Is E(F(M)M) ⊆ F(M)D(X) (Exercises 4.3.54 and 4.3.56)? Here, M is an archime- dean `-group, F(M) is its f -ring of f -endomorphisms, X is the Stone space of its Boolean algebra of polars, and E(F(M)M) denotes its injective hull as an F(M)-module. 11. Determine F(M) when M is a free abelian `-group (Theorems 4.5.6 and 4.5.10). 609 610 Open Problems 12. If an f -algebra over C is V-`-unitable for a suitable variety of `-algebras V , must it be a C-unitable f -algebra (Theorem 3.4.2 and remarks after it)? For example, V is the variety of almost f -algebras or the variety of sp-` algebras. 13. Conjecture: A regular sp-`-ring is an f -ring. (True if 1 2 R by Theorem 3.7.3). 14. Is a unital right self-injective f -ring necessarily left self-injective (Theorems 4.3.22 and 4.3.30)? 15. Is the maximal right quotient ring Q of the right nonsingular right f -ring R an `-ring extension of R provided QR is an f -module extension of RR? 16. If an W- f -group G has the property that each of its values is normal in its cover must each of its W-values be normal in its cover (Exercises 2.5.2, 2.5.18, and 2.5.19)? See [D] for descriptions of normal valued `-groups. 17. Develop the theory of f -modules over the f -ring C(X) where X is a topological space and over the f -ring D(X) where X is a Stone space or just an extremally disconnected space. 18. Is a free nonsingular R- f -module `-torsionless for R =C(X) or R = D(X) where X is as in the previous problem (Theorems 4.5.6 and 4.5.10)? 19. Determine the nonsingular ℵa -injective right f -modules over a semiprime right q f -ring (Theorem 4.4.6). 20. Does the category of unital right f -modules over a unital f -ring have nonzero injectives (Theorem 4.4.1)? The answer is no when the ring is an irredundant semiprime right q f -ring. 21. Conjecture: The polynomial ring D[x] over the totally ordered division ring D does not have an sp-lattice order extending that of D in which deg x+ ¸ 3 (Theorem 6.1.19). 22. Identify which fields, necessarily real algebraic extensions of the rationals, are O¤-fields (Theorems 6.2.13 and 6.2.14). 23. Is a free sp-`-algebra or a free f -algebra or a free `-algebra `-semiprime (The- orems 6.3.18 and 6.3.19)? The same question for the free unital `-algebras. 24. Is a totally ordered division ring which satisfies the identity j[[x;y];z]j · x2 ^ y2 ^ z2 a field (Theorem 6.3.9 and 6.3.11)? 25. 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