A non-contact communication system was developed for a ventilator-assisted patient with Werdnig–Hoffmann disease who had lost all voluntary movements except for those of the eye. The system detects the extraocular movements and converts them to either a ‘yes’ signal (produced by one lateral eyeball movement) or a ‘no’ signal (produced by two successive lateral eyeball movements) using a video camera placed outside the patient's visual field. The patient is thus able to concentrate on performing a task without any intrusion from the detection system. Once the setting conditions of the device have been selected, there is no need for any resetting, as the patient is unable to move his body. In addition to playing television games, the child can use the device to select television channels, compose music, and learn written Japanese and Chinese characters. This seems to broaden the patient's daily world and promote mental development.

Let E be a non-singular projective curve of genus g ≥ 0, P the projective line and let F be the surface E×P. Then it is well known that a ruled surface F* which is birational to F is biregular to a surface which is obtained by successive elementary transformations from F (for the notion of an elementary transformation, see [3]).

In the present paper, we first prove the following

The fourteenth problem of Hilbert asked finite generation of a certain class of rings and had a counter-example (cf. [4]). On the other hand, many mathematicians gave various sufficient conditions for finite generation of such rings (see, for instance, [9], [5] and [8]). The purpose of the present paper is to give a new sufficient condition. The class of rings to be treated is much more general than those treated before, except for the one in [8].

There is a problem called the chain problem of prime ideals, which asks, when 0 is a Noetherian local integral domain, whether the length of an arbitrary maximal chain of prime ideals in 0 is equal to rank 0 or not.

The proof of Theorem 1 in the paper “On Krull’s conjecture concerning valuation rings” (vol. 4 (1952) of this journal) is not correct. We want to give here a corrected proof of the theorem: From p. 30, l. 14 to p. 31, l. 7 should be changed as follows.

Previously the following question was offered by Zariski [6]:

Is any normal Noetherian local ring analytically irreducible?

In the present note, we will construct a counter-example against the question.

TERMINOLOGY. A ring (integrity domain) means always a commutative ring (integrity domain) with identity. A normal ring is an integrity domain which is integrally closed in its field of quotients. When 0 is an integrity domain, the integral closure of 0 in its field of quotients is called the derived normal ring of 0.

In a previous paper, we studied a general theory of integrally closed Henselian integrity domains and some properties of Henselian valuation rings. The present paper is its continuation. The main aim of the present paper is to study a general theory of Henselian local integrity domains in the present paper we call a ring o a local ring if o is a quasi-local ring and if the intersection of all powers of the maximal ideal of o is zero, and in this case we introduce a topology by taking the system of all powers of the maximal ideal as a system of neighbourhoods of zero.

Recently one of the writers used, in proving a theorem on the commutativity of certain division rings, the following lemma :

I. Let L be a field and K be its proper subfield. Except either when L is of characteristic p ≠ O and absolutely algebraic or ivhen L is algebraic and purely inseparable over K, there exists a pair of distinct (special exponential) valuations in L which coincide on K.

II. Let K be a field which is either of characteristic 0 or not absolutely algebraic, and L be its separable finite extension. There exist then infinitely many valutions in L which are of 1st degree over K.

Previously C. Chevalley [1] proved the followings:

1. Let x1,…, Xn be algebraically independent elements over a field t which has infinitely many elements. Then :

a) If y is an element of [x1,…, xn] and if y is not in , then there exist elements y2 …, yn of [x1 ,…, xn ] such that [x1 ,…, xn ] is integral over [y1, y2 ,…, yn ]

The notion of Henselian rings was introduced by G. Azuma-ya [l]. We concern ourselves in the present paper mainly with Henselizations of integrally closed integrity domains. Chapter I deals with general integrally closed integrity domains. As a preparation of our studies, we introduce the notion of decomposition rings analogously as in the case of fields (§1). And then we define the notions of (local) Henselian rings and Henselizations of integrally closed integrity domains, and obtain several results concerning characterizations of Henselian rings and the uniqeness of Henselizations (§2).

The proof of Proposition 2 and that of Corollary to Proposition 3 in my previous paper “On the structure of complete local rings” are not correct Here we want to correct them.

Previously W. Krull conjectured that every completely integrally closed primary domain of integrity is a valuation ring, The main purpose of the present paper is to construct in §1 a counter example against this conjecture. In § 2 we show a necessary and sufficient condition that a field is a quotient field of a suitable completely integrally closed primary domain of integrity which is not a valuation ring.

The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ring R is complete if (and only if) R/p is complete for every minimal prime divisor p of zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.

In my previous paper “On the theory of semi-local rings,” we saw that if a semi-local ring R with maximal ideals p1 …,ph is a subdirect sum of local rings Rm,2) then R is the direct sum of R[p1] (proposition 15, (slr)1)) and that a complete semi-local ring is a direct sum of complete local rings (Remark to proposition 5, (slr)).

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.