In the theory of commutative rings and their ideals, you can find mayor and not surpassed applications in number theory, especially in factorisation of numbers in quadratic and higher number fields, where roots are adjunctions to rational numbers and real approximations are notneeded.In this text, you can find in 10 chapters homomorphisms, rings, and ideals in elementary algebra, fundamental algebra and examples of rings, polynomial rings, residue classes, modules and ideals, primes, prime elements, and factorisation, residue classes and ideals, special rings, ideals in number theory, and a short history of ideals. After the text, a choke of literature, the, until now, collected corrections, and an index can be found.