Ribenboim^ On ordered modules

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We consider the above determined limit ordinal ?y, and we show that Q^ is an Ji^^-o-injective module.

If iV, P are ordered ^-modules such that #(jP) < i<^, if i: iV-> jP is an o-mono- morphism, f:N->Q^ an o-homomorphism, then for every n^N we have f(n)^Q^, hence there exists the smallest ordinal i{n) < rj such that f(n) = [^к^)], for some

hn ) ^ Q^in)' Let f' = sup {^(n) I n € iV}; hence f(N) ^g^M')- Since #(iV) < K., then Г < ^, so I = I' + 1 < 7^ and I is not a limit ordinal. Since g^.^ is an o-monomorphism, then f:N-^Q^ may be written as f == g^,J' where fiN-^Q^. is defined as follows: f(n) is the unique element in Q^. such that f{n) = g^^rjifi^))- The mapping /' is an ö-homomorphism (since g^,^ is an o-monomorphism).

Since Q^ is an K«^-o-injective module, and #(i^) < K* < i^«,, there exists an ö-homomorphism Г : P-> Q^ such that Vt == /у.

0 We define Z = kJ^^V : P-^ Q^, Then I is an o-homomorphism such that

It = k^jjt = kjj.r = g,,^r = /.

Finally , Q^ is an o-essential extension of il/, because g_^^^: M-^Q^i^ an essential o-monomorphism. This proves the theorem.

We define now the concept of o-injective dimension.

Let Л be a directed ring. Let ilf be a semi-closed ordered ^-module, let # (M) < J^^; hence there exists an J^^-o-injective module Q such that Ж ^ Ç (as ordered submodule). Let \l^ß be the smallest cardinal number with the above property, and let Qq be an J^^^-o-injective module such that M^Qq. Let M^= QqK^Mq}, Repeating the same argument with M^ we obtain an JÇ^^-o-injective module Çi» such that M^ ^Qi» In this way, we have the strictly o-exact sequences

Combining these sequences, we obtain the strictly o-exact sequence 0_^ <M> =^ Q^Juq^Jl^Q^Jl^.,.. where /« = ИРо. /i = hPu ....

For example, if M is J$^-o-injective (for some ordinal ß) we may take Qq= M and obtain the strictly ö-exact sequence 0=^ M =^ M-^0,

We say that the ordered module M has o-injective dimension at most n when there exists a strictly o-exact sequence

where each Q^ is J^^j.-o-injective (for some ß^.

Journal für Mathematik. Band 225. 1Э