2024-04-14

531: %Ring Name% Module

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definition of %ring name% module

Topics


About: module

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of %ring name% module.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description


Here is the rules of Structured Description.

Entities:
R: ∈{ the rings }
∗M: ∈{ the sets } with any +:M×M→M (addition) operation and any .:R×M→M (scalar multiplication) operation
//

Conditions:
1) ∀m1,m2∈M(m1+m2∈M) (closed-ness under addition)
∧
2) ∀m1,m2∈M(m1+m2=m2+m1) (commutativity of addition)
∧
3) ∀m1,m2,m3∈M((m1+m2)+m3=m1+(m2+m3)) (associativity of additions)
∧
4) ∃0∈M(∀m∈M(m+0=m)) (existence of 0 element)
∧
5) ∀m∈M(∃m′∈M(m′+m=0)) (existence of inverse vector)
∧
6) ∀m∈M,∀r∈R(r.m∈M) (closed-ness under scalar multiplication)
∧
7) ∀m∈M,∀r1,r2∈R((r1+r2).m=r1.m+r2.m) (scalar multiplication distributability for scalars addition)
∧
8) ∀m1,m2∈M,∀r∈R(r.(m1+m2)=r.m1+r.m2) (scalar multiplication distributability for elements addition)
∧
9) ∀m∈M,∀r1,r2∈R((r1r2).m=r1.(r2.m)) (associativity of scalar multiplications)
∧
10) ∀m∈M(1.m=m) (identity of 1 multiplication)
//

2: Natural Language Description


Any set, M, with any +:M×M→M (addition) operation and any .:R×M→M (scaler multiplication) operation with respect to any ring, R, that satisfies these conditions: 1) for any elements, m1,m2∈M, m1+m2∈M (closed-ness under addition); 2) for any elements, m1,m2∈M, m1+m2=m2+m1 (commutativity of addition); 3) for any elements, m1,m2,m3∈M, (m1+m2)+m3=m1+(m2+m3) (associativity of additions); 4) there is a 0 element, 0∈M, such that for each m∈M, m+0=m (existence of 0 element); 5) for any element, m∈M, there is an inverse element, m′∈M, such that m′+m=0 (existence of inverse vector); 6) for any element, m∈M, and any scalar, r∈R, r.m∈M (closed-ness under scalar multiplication); 7) for any element, m∈M, and any scalars, r1,r2∈R, (r1+r2).m=r1.m+r2.m (scalar multiplication distributability for scalars addition); 8) for any elements, m1,m2∈M, and any scalar, r∈R, r.(m1+m2)=r.m1+r.m2 (scalar multiplication distributability for elements addition); 9) for any element, m∈M, and any scalars, r1,r2∈R, (r1r2).m=r1.(r2.m) (associativity of scalar multiplications); 10) for any element, m∈M, 1.m=m (identity of 1 multiplication)

3: Note


. is often omitted in notations like rm instead of r.m.

The requirements 1) ~ 10) are in fact parallel to those for vectors space; the difference between vectors space and module is only that the scalar structure is a field or a ring, which makes a significant difference, because for example, for a module, r1m1+r2m2+r3m3=0 where r1≠0 does not imply that m1 is a linear combination of m2 and m3, because r1−1 is not guaranteed to exist.

References
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