ДИСКРЕТНАЯ МАТЕМАТИКА Семинар 10. КОЛЬЦА. ПОЛЯ

Òêà÷åâ Ñ.Á.
êàô. Ìàòåìàòè÷åñêîãî ìîäåëèðîâàíèÿ
ÌÃÒÓ èì. Í.Ý. Áàóìàíà
ÄÈÑÊÐÅÒÍÀß ÌÀÒÅÌÀÒÈÊÀ
ÈÓ5 | 4 ñåìåñòð, 2015 ã.
Ñåìèíàð 10. ÊÎËÜÖÀ. ÏÎËß.
ÐÅØÅÍÈÅ ÑËÀÓ
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1. Êîëüöà.
Îïðåäåëåíèå 10.1. Êîëüöî | ýòî àëãåáðà ñ äâóìÿ áèíàðíûìè è äâóìÿ íóëüàðíûìè îïåðàöèÿìè
R = (R, +, ·, 0, 1)
òàêàÿ, ÷òî:
1) àëãåáðà (R, +, 0) | êîììóòàòèâíàÿ ãðóïïà;
2) àëãåáðà (R, ·, 1) | ìîíîèä ;
3) èìååò ìåñòî äèñòðèáóòèâíîñòü îïåðàöèè · (óìíîæåíèÿ
êîëüöà) îòíîñèòåëüíî îïåðàöèè + (ñëîæåíèÿ êîëüöà):
a · (b + c) = a · b + a · c,
(b + c) · a = b · a + c · a.
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Îïåðàöèþ + íàçûâàþò ñëîæåíèåì êîëüöà, · | óìíîæåíèåì êîëüöà,ýëåìåíò 0 | íóëåì êîëüöà, ýëåìåíò 1 |
åäèíèöåé êîëüöà.
Îïðåäåëåíèå 10.2. Êîëüöî íàçûâàþò êîììóòàòèâíûì,
åñëè îïåðàöèÿ óìíîæåíèÿ â íåì êîììóòàòèâíà.
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Ïðèìåð 1.
à) Àëãåáðà (Z, +, ·, 0, 1) åñòü êîììóòàòèâíîå êîëüöî.
á) Àëãåáðà (N ∪ {0}, +, ·, 0, 1) êîëüöîì íå áóäåò, ïîñêîëüêó
(N ∪ {0}, +) | êîììóòàòèâíûé ìîíîèä, íî íå ãðóïïà.
á) Àëãåáðà
Zk = ({0, 1, 2, . . . , k − 1}, ⊕k , k , 0, 1)
(ïðè k ≥ 1 ), åñòü êîììóòàòèâíîå êîëüöî.
Åãî íàçûâàþò êîëüöîì âû÷åòîâ ïî ìîäóëþ k .
Àääèòèâíàÿ ãðóïïà êîëüöà åñòü àääèòèâíàÿ ãðóïïà âû÷åòîâ
ïî ìîäóëþ k ,
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Îïðåäåëåíèå 10.3. Íåíóëåâûå ýëåìåíòû a è b êîëüöà R
íàçûâàþò äåëèòåëÿìè íóëÿ, åñëè a · b = 0 .
Çàäà÷à 4. Ñóùåñòâóþò ëè äåëèòåëè íóëÿ â êîëüöå âû÷åòîâ
ïî ìîäóëþ 4 Z4 .
 êîëüöå Z5 ?
Ïðè êàêèõ n Zn íå ñîäåðæèò äåëèòåëåé íóëÿ?
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2. Ïîëÿ
Îïðåäåëåíèå 10.4. Êîëüöî, â êîòîðîì ìíîæåñòâî âñåõ íåíóëåâûõ ýëåìåíòîâ ïî óìíîæåíèþ îáðàçóåò ãðóïïó, íàçûâàþò
òåëîì.
Êîììóòàòèâíîå òåëî íàçûâàþò ïîëåì.
Ãðóïïó íåíóëåâûõ ýëåìåíòîâ ïîëÿ ïî óìíîæåíèþ íàçûâàþò
ìóëüòèïëèêàòèâíîé ãðóïïîé ýòîãî ïîëÿ.
Ïðèìåð 2.
à) Àëãåáðà (Q, +, ·, 0, 1) åñòü ïîëå, íàçûâàåìîå ïîëåì
ðàöèîíàëüíûõ ÷èñåë.
á) Àëãåáðà (R, +, ·, 0, 1) åñòü ïîëå, íàçûâàåìîå ïîëåì
âåùåñòâåííûõ ÷èñåë.
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Çàäà÷à 6.1. Êàêèå èç ÷èñëîâûõ ìíîæåñòâ îáðàçóþò êîëüöî
îòíîñèòåëüíî îáû÷íûõ îïåðàöèé óìíîæåíèÿ è ñëîæåíèÿ:
(à) ìíîæåñòâî íåîòðèöàòåëüíûõ√öåëûõ ÷èñåë;
(á) ìíîæåñòâî ÷èñåë âèäà x + 2y , x, y ∈ Q ?
Êàêèå èç óêàçàííûõ êîëåö ÿâëÿþòñÿ ïîëÿìè?
Çàäà÷à 6.2. Êàêèå èç ìíîæåñòâ ìàòðèö îáðàçóþò êîëüöî
îòíîñèòåëüíî ìàòðè÷íûõ îïåðàöèé óìíîæåíèÿ è ñëîæåíèÿ?
Êàêèå èç êîëåö ÿâëÿþòñÿ ïîëÿìè?
(à) ìíîæåñòâî ìàòðèö âèäà
(á) ìíîæåñòâî ìàòðèö âèäà
a b
0 c
, a, b, c ∈ R ?
a b
−b a
, a, b ∈ R ?
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Òåîðåìà 1. Â ëþáîì êîëüöå âûïîëíÿþòñÿ ñëåäóþùèå
òîæäåñòâà
1) a · 0 = 0 · a = 0 .
2) (a − b) · c = a · c − b · c ,
c · (a − b) = x · a − c · b , ãäå ðàçíîñòü a − b åñòü ïî
îïðåäåëåíèþ a − b = a + (−b) .
Ñëåäñòâèå 10.1. Â ëþáîì êîëüöå ñïðàâåäëèâû òîæäåñòâà:
a · (−b) = (−a) · b = −a · b
(â ÷àñòíîñòè, (−1) · x = x · (−1) = −x ).
Òàêèì îáðàçîì, ïðîèçâîäÿ âû÷èñëåíèÿ â ëþáîì êîëüöå
(ïîëå), ìîæíî ðàñêðûâàòü ñêîáêè è ìåíÿòü çíàêè òàê æå,
êàê â îáû÷íîé øêîëüíîé àëãåáðå.
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Çàäà÷à 6.3.
Ðåøèòü â ïîëå Z3 è â ïîëå Z5 ñèñòåìó óðàâíåíèé:

 x + 2y = 1,
y + 2z = 2,
 2x + z = 1.
Çàäà÷à 6.4.
Ðåøèòü â ïîëå Z5 è â ïîëå Z7 ñèñòåìó óðàâíåíèé:
2x + 3y = 1,
3x − 4y = 2.
Çàäà÷à 6.5.
Ðåøèòü â ïîëå Z7 ñèñòåìó óðàâíåíèé:

 3x + 4y + 5z = 2
3x + 2y + 3z = 4,
 x + y + 4z = 2.
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Çàäà÷à 6.6. Óñòàíîâèòü, èìååò ëè ðåøåíèå â ïîëå Z11
ñèñòåìà óðàâíåíèé:

 3x + 7y + 10z = 2
5x + 2y + 8z = 4,
 9x + 3y + 7z = 6.
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Äîìàøíåå çàäàíèå
Çàäà÷à Ä6.1.
Ðàçðåøèìà ëè â êîëüöå Z21 ñèñòåìà óðàâíåíèé:
5x + 2y = 1,
y − 11x = 13?
Çàäà÷à Ä6.2. Óñòàíîâèòü, èìååò ëè ðåøåíèå â ïîëå Z11
ñèñòåìà óðàâíåíèé:

 3x + 7y + 10z = 2
5x + 2y + 8z = 4,
 9x + 3y + 7z = 6.
Åñëè ðåøåíèå íå åäèíñòâåííî, îïèñàòü ìíîæåñòâî ðåøåíèé.
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Çàäà÷à Ä6.3. Óñòàíîâèòü, èìååò ëè ðåøåíèå â ïîëå Z11
ñèñòåìà óðàâíåíèé:

 3x + 4y + 5z = 6
6x + 2y + 8z = 2,
 9x + 1y + 4z = 7.
Åñëè ðåøåíèå íå åäèíñòâåííî, îïèñàòü ìíîæåñòâî ðåøåíèé.
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Äîïîëíèòåëüíûå çàäà÷è
10.1. Êîëüöî R íàçûâàåòñÿ áóëåâûì, åñëè ∀x ∈ R
x2 = x . Äîêàçàòü:
(à) â ëþáîì áóëåâîì êîëüöå ∀x ∈ R x + x = 0 ;
(á) ëþáîå áóëåâî êîëüöî êîììóòàòèâíî;
(â) â ëþáîì áóëåâîì êîëüöå ìîùíîñòè áîëüøå 2 åñòü
äåëèòåëè íóëÿ.
10.2. Äîêàçàòü, ÷òî (2M , 4, ∩, ∅, M ) | áóëåâî êîëüöî.
Äîêàçàòü, ÷òî îíî èçîìîðôíî Z2 ïðè |M | = 1 .
10.3. Áóäåò ëè ëþáîå êîëüöî Z2n , n ≥ 1 , áóëåâûì?
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