Linguistic Premise Premise of Algebraic Linguistics 3-3 / September 24, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 3-3

　　  TANAKA Akio

13 <tensor product>

A module M, N

Tensor product     M ⊗_A N

Given conditions

(1) M ⊗_A N is A module that is generated by { x ⊗ y | x ∈M, y ∈ N }

(2) ( x + x’ ) ⊗ y = x ⊗ y + x’ ⊗ y,   x ⊗ (y + y’ ) = x ⊗ y + x ⊗ y’     a ∈ A   a ( x ⊗ y ) = ax ⊗y = x ⊗ ay

14 <multiplicative set>

Ring     A

Subset of A     S

1 ∈ S, 0 ∉ S

S is closed by multiplication.    

S is Multiplicative set.

15 <localization>

Ring     A

S is A multiplicative set,

S^-1A = { a/s | a ∈ A, s ∈ S }

S^-1A is ring.

16 <total quotient ring, canonical map>

Ring     A

Ring     S = { s ∈ A | s is non-zero divisor }

S^-1A is total quotient ring.

A is subring of total quotient ring S^-1A.

Canonical is next..

(1) Map i_A : A → S^-1A,   i_A (a ）= a/1

(2) Map i_M : M → S^-1M,   I (x ) = x/1

Tokyo September 24, 2007

Sekinan Research Field of Language