Constructions of Commutative Semirings and Radical Rings

In this dissertation we deal with constructive methods applied to the commutative semirings and commutative radical rings. In Chapter 2 we study the class S of the commutative subdirectly irreducible radical rings. We present a few constructive methods for them and using the reflection of the category of the commutative rings into the category of the commutative radical rings we derive a lot of examples of rings in S with various properties. We prove that a ring S 2 S is noetherian if and only if it is finite. We show partial results in the classification of factors of S modulo monoliths. In Chapter 3 we introduce, using the p-prime valuation for all primes p, a set of characteristic sequences that can be assign to every subsemiring of Q+. We find and classify all maximal subsemirings of positive rational numbers and show that every proper subsemiring of Q+ is contained in at least one of them. This results was published in [16]. In Chapter 4 we construct, using the approach from the Chapter 4, a new large subclass of the class CongSimp of all proper congruence-simple subsemirings of Q+, classify all the maximal elements of CongSimp and show that every element of CongSimp is contained in at least one of them. In Chapter 5 we find an equivalent condition under which is the semiring Q+[ ] C, 2 C, contained in...