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Because the subsequent instance exhibits, the answer's no. instance four. 6 As a module over itself, is unfastened at the set { ¸¹. For any € , the set { ~ ¸' “ '  {¹ is a unfastened cyclic submodule of {, however the quotient {- module {°{ is isom orphic to { through the map ²" b {³ ~ " mod and because { isn't loose as a {-module, nor is {°{. The Correspondence and Isomorphism Theorems The correspondence and isomorphism theorems for vector areas have analogs for modules. Theorem four. eight (The correspondence theorem) enable : be a submodule of four . Then the functionality that assigns to every intermediate submodule : ‹ ; ‹ four the quotient submodule ; °: of 4°: is an order-preserving with ( recognize to set inclusion) one-to-one correspondence among submodules of four containing : and all submodules of 4°: . Theorem four. nine (The ) first isomorphism theorem allow ¢ four ¦ five be an nine - homomorphism. Then the map Z¢ 4°ker²³ ¦ five outlined via Z²# b ² ker ³³ ~ # is an nine -embedding and so four š ² im ³ ker²³ Modules I: easy homes 119 Theorem four. 10 The ( moment isomorphism theorem) permit four be an nine -module and enable : and ; be submodules of four . Then : b ; : š ; : q ; Theorem four. eleven The ( 3rd isomorphism theorem) permit four be an nine -module and consider that : ‹ ; are submodules of four . Then four °: four š ; °: ; Direct Sums and Direct Summands The definition of direct sum of a family members of submodules is an instantaneous analog of the definition for vector areas. Definition The of exterior direct sum nine -modules fourá à Á four , denoted by way of four ~ four ^ Ä ^ four is the -module whose parts are ordered -tuples four ~ ¸²#Á à Á #³ “ #  fourá ~ Á à Á ¹ with componentwise operations ²"Á à Á "³ b ²#Á à Á #³ ~ ²" b #Á à Á " b #³ and ²#Á à Á #³ ~ ²#Á à Á #³ for .  nine We depart it to the reader to formulate the definition of exterior direct sums and items for arbitrary households of modules, in direct analogy with the case of vector areas. Definition An nine -module four is the ( ) inner of a kinfolk direct sum < ~ ¸: “  zero¹ of submodules of four , written four ~ < or four ~ : 0 if the next carry: 1) (Join of the relations) four is the sum ( sign up for) of the relations < : = ~ : 0 a hundred and twenty complicated Linear Algebra 2) (Independence of the kinfolk) for every  zero , p s : q : ~ ¸¹ q £ t for this reason, each one : is named a right away summand of four . If < ~ ¸:Á à Á :¹ is a finite kin, the direct sum is usually written four ~ : l Ä l : eventually, if four ~ : l ; , then : is expounded to be complemented and ; is termed a supplement of : in four . As with vector areas, now we have the next important characterization of direct sums. Theorem four. 12 enable < ~ ¸: “  zero¹ be a relations of targeted submodules of an nine - module four . the next are an identical: 1) (Independence of the kin) for every  zero , p s : q : ~ ¸¹ q £ t 2) (Uniqueness of expression for ) The 0 point can't be written as a sum of nonzero parts from specified submodules in < . three) ( ) forte of expression each nonzero #  four has a different, with the exception of order of phrases, expression as a sum # ~ b Ä b of nonzero parts from targeted submodules in < .