TY - JOUR
T1 - On the existence and non-existence of complementary radical and semisimple classes
AU - Veldsman, S.
AU - Wiegandt, R.
PY - 1984
Y1 - 1984
N2 - In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.
AB - In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.
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U2 - 10.1080/16073606.1984.9632332
DO - 10.1080/16073606.1984.9632332
M3 - Article
AN - SCOPUS:84952196889
SN - 1607-3606
VL - 7
SP - 213
EP - 224
JO - Quaestiones Mathematicae
JF - Quaestiones Mathematicae
IS - 3
ER -