The basic concepts of algebra, such as monoids and groups, rings and fields, as well as modules and algebras, are discussed. Homomorphisms play an early decisive role as structure-preserving mappings and lead to standard constructions such as quotient formations, sums and products, and the discussion of free objects. Universal properties of the constructed objects are highlighted and prepare for a later more category-theoretical view. Natural compatibility conditions for operations motivate the introduced concepts. The starting point is monoids and their unit groups. The theory of groups is carried out up to the Sylow theorems and concretized on finite permutation groups. Application examples include the quadratic reciprocity law and the Pólya enumeration formula. Rings, modules, and algebras are structures with multiple canonically connected operations. The residue class rings of are the minimal rings and essential objects of elementary number theory. With the general Chinese remainder theorem, their structure and that of their unit groups, the prime residue class groups, are clarified. Modules and vector spaces are treated, including the concept of rank and dimension. The section on algebras discusses, among other things, very detailed (also non-commutative) polynomial algebras up to the Hilbert basis theorem and the prime factor decomposition. Principal ideal domains and in particular Euclidean domains find their place. Application examples include finite fields and the two and four squares theorem.