By M. Petrich
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Additional info for Categories of Algebraic Systems: Vector and Projective Spaces, Semigroups, Rings and Lattices
Fuchs, Abelian groups, Pergamon Press, 1960. 46, line 2 [=]: For ‘‘a finite set’’ read ‘‘the finite set S ’’. 5 [~]: Lang writes ‘‘mA is free, hence A is free’’. The argument is that multiplication by m is a homomorphism A → mA; it is clearly onto, and it is one-to-one because A is torsion-free. Hence A ∼ = mA, so if the latter is free, the former is, also. 9. The dual group. 24-25, the ‘‘standard’’ cyclic group of order n was written Z ⁄ n Z; here Lang writes Zn for an arbitrary cyclic group of order n.
38 we mean the universal tuple, then these symbols denote its first component, and the whole tuple will have the form (A × B, f, g) or ( I I Ai , ( fi )). , or π A etc. ) The same comments apply to the symbol I I for ‘‘coproduct’’, introduced on the next page. 60, last Example [<]: Ignore this unless you have seen tensor products of rings. 61, three lines following the triangular diagram [=]: Replace these lines by: ‘‘The product in Z of two objects f : X → Z and g : Y → Z is called the fibered product of X and Y (or of f and g) over Z, and is denoted X × Z Y.
I leave it to you which way you prefer to adjust your conventions to make this consistent with our non-categorical notation; it will not affect the material to follow. 55 beginning ‘‘Next we consider ... ’’ and ‘‘More generally ... ’’. ) Most of Lang’s other examples involve material that few beginning students will have had, and are not clearly presented. ) In place of (or in addition to) Lang’s examples, read the examples and (optionally) the notes on category theory and set theory below, then go back to Lang where indicated at the end of those notes.
Categories of Algebraic Systems: Vector and Projective Spaces, Semigroups, Rings and Lattices by M. Petrich