1 <finitely generated>
Group G
Subset of G S
G is generated by finite set S. G is finitely generated.
2 <ascending chain rule>
Commutative ring A
Ideal of A
Ascending chain of I stops finitely.
The situation satisfies ascending chain rule.
3 <descending chain rule>
Commutative ring A
Ideal of A
Descending chain of I stops finitely.
The situation satisfies descending chain rule.
4 <maximum element>
Set defined by order X
Element of X a, x
x that is a < x does not exist.
a is maximum element.
5 <minimum element>
Set defined by order X
Element of X b, x
x that is b > x does not exist.
b is minimum element.
4 <Noetherian ring>
Commutative ring A that satisfies next equivalent conditions is Noetherian ring.
(1) A satisfies ascending chain rule on ideal.
(2) Ideal family of A has maximum element.
(3) Ideal of A is finitely generated.
5 <Artinian ring>
Commutative ring A that satisfies next equivalent conditions is Artininian ring.
(1) A satisfies descending chain rule on ideal.
(2) Ideal family of A has minimum element.
6 <module>
Additive group M
Ring A
M that has action of A is A module.
M satisfies next conditions.
(1) a ( x + y ) = ax + ay, ( a + b ) x = ax + bx
(2) a ( bx ) = ( ab ) x, 1x = x
7 <direct sum>
A module M, N
Structure of A module is given by set of product M ×N. The situation is expressed by M ⊕ N.
Direct sum of n-M, i.e. M ⊕M ⊕….⊕M is expressed by Mn.
Tokyo September 23, 2007
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