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Lecturer(s)
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Kühr Jan, prof. RNDr. Ph.D.
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Chajda Ivan, prof. RNDr. DrSc.
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Kurač Zbyněk, Mgr.
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Course content
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1. Binary relations, reflexive, symmetric and transitive relations. Equivalence relations and partitions, quotient sets. 2. Grupoids, semigroups and groups. Natural and integer powers in semigroups and groups. Homomorphisms and congruence relations, quotient grupoids, the homomorphism theorem for grupoids. Subgroups and normal subgroups of groups. Congruence relations and homomorphisms of groups. Quotient groups. The homomorphism theorem for groups, isomorphism theorems. Subgroup generated by a set, order of a subgroup and of an element. Cyclic groups. Permutation groups, the Cayley theorem. Direct products of groupoids. 3. Rings, integral domains and fields. Ideals, prime ideals and maximal ideals. Homomorphism and congruence relations, quotient rings. The homomorphism theorem. Order of an element, characteristic of a ring. 4. Divisibility in integral domains. Units, irreducible and prime elements. Greatest common divisor, least common multiple. Ideal generated by a set, pricipal ideal domains. Euclidean domains, Gaussian domains. Direct products of rings.
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Learning activities and teaching methods
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Lecture, Monologic Lecture(Interpretation, Training)
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Learning outcomes
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To understand the rudiments of the theory of groups and rings.
1. Knowledge Define basic notions, describe basic constructions and recall fundamental theorems of theory of groups and rings.
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Prerequisites
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unspecified
KAG/ALG2 and KAG/ALG1
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Assessment methods and criteria
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Oral exam, Written exam
Credit: attendance at seminars, written test. Exam: oral exam, students have to demonstrate their knowledge and understanding of the subject matter.
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Recommended literature
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Bican L. (2004). Lineární algebra a geometrie. Academia Praha.
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Halaš R., Chajda I. (1999). Cvičení z algebry. VUP Olomouc.
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I., Chajda. (1999). Úvod do algebry. UP Olomouc.
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Jukl M. (2006). Lineární algebra. UP Olomouc.
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Krutský F. (1995). Algebra I.. VUP Olomouc.
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Rachůnek, J. (2005). Grupy a okruhy. VUP Olomouc.
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Stanovský D. (2010). Základy algebry. Matfyzpress Praha.
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