By Friedrich Kasch, Adolf Mader

This publication bargains for the 1st time a entire and readable exposition of effects at the overall. even though facing fresh study, the cloth is offered to someone with a easy wisdom of ring and module concept. a quick creation to torsion-free Abelian teams is integrated . the topic is on no account exhausted and subject matters for extra study can simply be discovered.

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A⊕ i∈I Qi ⊆∗ V. 3. Further properties 47 2) Assume that W is lp and let B ⊆ W . Then there exists a maximal independent set {Pi | i ∈ I} of projective modules Pi ⊆⊕ W , Pi ⊆ B with the following property: If C ⊆ B and i∈I Pi ∩ C = 0, then C ⊆◦ W . This means that P := i∈I (1) Pi is “nearly large” in B. Proof. 1) Since the union of an ascending chain of independent sets {Qi | i ∈ I} of injective submodules Qi ⊆ V with A∩ i∈I Qi = 0 is again such a set, we can apply Zorn’s Lemma. Therefore we can assume that {Qi | i ∈ I} is maximal.

I) f induces isomorphisms fn : Rn x → f (x) ∈ Tn , ii) f (Sn ) ⊆ Un , Ker(f ) ⊆ Sn , iii) Rn Rn+1 , Sn Sn+1 , Tn+1 Tn , Un+1 Un . We are now ready to give an example of a non–regular ring R with Tot(R) = 0. 5. There exist rings with trivial total that are not regular. Proof. Let S := QN with component–wise addition and multiplication. Denote the elements of S by (qi ). We consider the subring R of S consisting of all elements (qi ) for which there is an integer n (depending on (qi )) with qi ∈ Z for all i ≥ n.

Then for every B ⊆ W , there exists a projective direct summand P ⊆⊕ W and U ⊆◦ W such that P ∩ U = 0, B = P ⊕ U. In particular, for B = W it follows that P = W is projective. Proof. 1) Let Q be an injective submodule that is maximal with respect to A∩Q = 0. Assume that A ⊕ Q is not large in V . Then there exists an injective submodule Q0 ⊆ V with (A ⊕ Q) ∩ Q0 = 0, and then Q + Q0 = Q ⊕ Q0 is again injective and A ∩ (Q ⊕ Q0 ) = 0. This contradicts the maximality of Q, hence A ⊕ Q ⊆∗ V . If A = 0, then Q ⊆∗ V ; but, because of G ⊆⊕ V , this is possible only for Q = V .