:: RING_1 semantic presentation
theorem Th1: :: RING_1:1
theorem Th2: :: RING_1:2
theorem Th3: :: RING_1:3
:: deftheorem Def1 defines quasi-prime RING_1:def 1 :
:: deftheorem Def2 defines prime RING_1:def 2 :
:: deftheorem Def3 defines quasi-maximal RING_1:def 3 :
:: deftheorem Def4 defines maximal RING_1:def 4 :
theorem Th4: :: RING_1:4
Lemma8:
for b1 being Ring
for b2 being Ideal of b1ex b3 being Equivalence_Relation of the carrier of b1 st
for b4, b5 being set holds
( [b4,b5] in b3 iff ( b4 in the carrier of b1 & b5 in the carrier of b1 & ex b6, b7 being Element of b1 st
( b6 = b4 & b7 = b5 & b6 - b7 in b2 ) ) )
definition
let c1 be
Ring;
let c2 be
Ideal of
c1;
func EqRel c1,
c2 -> Relation of
a1 means :
Def5:
:: RING_1:def 5
for
b1,
b2 being
Element of
a1 holds
(
[b1,b2] in a3 iff
b1 - b2 in a2 );
existence
ex b1 being Relation of c1 st
for b2, b3 being Element of c1 holds
( [b2,b3] in b1 iff b2 - b3 in c2 )
uniqueness
for b1, b2 being Relation of c1 st ( for b3, b4 being Element of c1 holds
( [b3,b4] in b1 iff b3 - b4 in c2 ) ) & ( for b3, b4 being Element of c1 holds
( [b3,b4] in b2 iff b3 - b4 in c2 ) ) holds
b1 = b2
end;
:: deftheorem Def5 defines EqRel RING_1:def 5 :
theorem Th5: :: RING_1:5
theorem Th6: :: RING_1:6
theorem Th7: :: RING_1:7
theorem Th8: :: RING_1:8
theorem Th9: :: RING_1:9
theorem Th10: :: RING_1:10
definition
let c1 be
Ring;
let c2 be
Ideal of
c1;
func QuotientRing c1,
c2 -> strict doubleLoopStr means :
Def6:
:: RING_1:def 6
( the
carrier of
a3 = Class (EqRel a1,a2) & the
unity of
a3 = Class (EqRel a1,a2),
(1. a1) & the
Zero of
a3 = Class (EqRel a1,a2),
(0. a1) & ( for
b1,
b2 being
Element of
a3ex
b3,
b4 being
Element of
a1 st
(
b1 = Class (EqRel a1,a2),
b3 &
b2 = Class (EqRel a1,a2),
b4 & the
add of
a3 . b1,
b2 = Class (EqRel a1,a2),
(b3 + b4) ) ) & ( for
b1,
b2 being
Element of
a3ex
b3,
b4 being
Element of
a1 st
(
b1 = Class (EqRel a1,a2),
b3 &
b2 = Class (EqRel a1,a2),
b4 & the
mult of
a3 . b1,
b2 = Class (EqRel a1,a2),
(b3 * b4) ) ) );
existence
ex b1 being strict doubleLoopStr st
( the carrier of b1 = Class (EqRel c1,c2) & the unity of b1 = Class (EqRel c1,c2),(1. c1) & the Zero of b1 = Class (EqRel c1,c2),(0. c1) & ( for b2, b3 being Element of b1ex b4, b5 being Element of c1 st
( b2 = Class (EqRel c1,c2),b4 & b3 = Class (EqRel c1,c2),b5 & the add of b1 . b2,b3 = Class (EqRel c1,c2),(b4 + b5) ) ) & ( for b2, b3 being Element of b1ex b4, b5 being Element of c1 st
( b2 = Class (EqRel c1,c2),b4 & b3 = Class (EqRel c1,c2),b5 & the mult of b1 . b2,b3 = Class (EqRel c1,c2),(b4 * b5) ) ) )
uniqueness
for b1, b2 being strict doubleLoopStr st the carrier of b1 = Class (EqRel c1,c2) & the unity of b1 = Class (EqRel c1,c2),(1. c1) & the Zero of b1 = Class (EqRel c1,c2),(0. c1) & ( for b3, b4 being Element of b1ex b5, b6 being Element of c1 st
( b3 = Class (EqRel c1,c2),b5 & b4 = Class (EqRel c1,c2),b6 & the add of b1 . b3,b4 = Class (EqRel c1,c2),(b5 + b6) ) ) & ( for b3, b4 being Element of b1ex b5, b6 being Element of c1 st
( b3 = Class (EqRel c1,c2),b5 & b4 = Class (EqRel c1,c2),b6 & the mult of b1 . b3,b4 = Class (EqRel c1,c2),(b5 * b6) ) ) & the carrier of b2 = Class (EqRel c1,c2) & the unity of b2 = Class (EqRel c1,c2),(1. c1) & the Zero of b2 = Class (EqRel c1,c2),(0. c1) & ( for b3, b4 being Element of b2ex b5, b6 being Element of c1 st
( b3 = Class (EqRel c1,c2),b5 & b4 = Class (EqRel c1,c2),b6 & the add of b2 . b3,b4 = Class (EqRel c1,c2),(b5 + b6) ) ) & ( for b3, b4 being Element of b2ex b5, b6 being Element of c1 st
( b3 = Class (EqRel c1,c2),b5 & b4 = Class (EqRel c1,c2),b6 & the mult of b2 . b3,b4 = Class (EqRel c1,c2),(b5 * b6) ) ) holds
b1 = b2
end;
:: deftheorem Def6 defines QuotientRing RING_1:def 6 :
for
b1 being
Ringfor
b2 being
Ideal of
b1for
b3 being
strict doubleLoopStr holds
(
b3 = QuotientRing b1,
b2 iff ( the
carrier of
b3 = Class (EqRel b1,b2) & the
unity of
b3 = Class (EqRel b1,b2),
(1. b1) & the
Zero of
b3 = Class (EqRel b1,b2),
(0. b1) & ( for
b4,
b5 being
Element of
b3ex
b6,
b7 being
Element of
b1 st
(
b4 = Class (EqRel b1,b2),
b6 &
b5 = Class (EqRel b1,b2),
b7 & the
add of
b3 . b4,
b5 = Class (EqRel b1,b2),
(b6 + b7) ) ) & ( for
b4,
b5 being
Element of
b3ex
b6,
b7 being
Element of
b1 st
(
b4 = Class (EqRel b1,b2),
b6 &
b5 = Class (EqRel b1,b2),
b7 & the
mult of
b3 . b4,
b5 = Class (EqRel b1,b2),
(b6 * b7) ) ) ) );
theorem Th11: :: RING_1:11
theorem Th12: :: RING_1:12
theorem Th13: :: RING_1:13
theorem Th14: :: RING_1:14
E19:
now
let c1 be
Ring;
let c2 be
Ideal of
c1;
set c3 =
EqRel c1,
c2;
let c4 be
Element of
(c1 / c2);
assume E20:
c4 = Class (EqRel c1,c2),
(1. c1)
;
let c5 be
Element of
(c1 / c2);
consider c6 being
Element of
c1 such that E21:
c4 = Class (EqRel c1,c2),
c6
by Th11;
consider c7 being
Element of
c1 such that E22:
c5 = Class (EqRel c1,c2),
c7
by Th11;
E23:
c6 - (1. c1) in c2
by E20, E21, Th6;
then E24:
c7 * (c6 - (1. c1)) in c2
by IDEAL_1:def 2;
E25:
c7 * (c6 - (1. c1)) =
(c7 * c6) - (c7 * (1. c1))
by VECTSP_1:43
.=
(c7 * c6) - c7
by VECTSP_1:def 13
;
thus c5 * c4 =
Class (EqRel c1,c2),
(c7 * c6)
by E21, E22, Th14
.=
c5
by E22, E24, E25, Th6
;
E26:
(c6 - (1. c1)) * c7 in c2
by E23, IDEAL_1:def 3;
E27:
(c6 - (1. c1)) * c7 =
(c6 * c7) - ((1. c1) * c7)
by VECTSP_1:45
.=
(c6 * c7) - c7
by VECTSP_1:def 19
;
thus c4 * c5 =
Class (EqRel c1,c2),
(c6 * c7)
by E21, E22, Th14
.=
c5
by E22, E27, E26, Th6
;
end;
theorem Th15: :: RING_1:15
theorem Th16: :: RING_1:16
theorem Th17: :: RING_1:17
theorem Th18: :: RING_1:18
theorem Th19: :: RING_1:19
theorem Th20: :: RING_1:20
theorem Th21: :: RING_1:21