Gần 30 năm viết và nói tiếng mẹ đẻ mà vẫn sai như thường: Anh sẽ chia “xẻ” với em (Em sẽ thành vài mảnh mỗi mảnh …)  !!!! Thế mà lại dám bàn về International Language. Thôi kệ collect những lỗi hay mắc phải, trong đó có cả lỗi của mình để tự nhắc mình không gặp phải nữa.

1. Let $R$ denotes a commutative ring –> Let $R$ denote a commutative ring.

Let k is integer –> Let k to be integer.

(Let is the imperative of the verb to let and has to be followed by an infinitive (without to). You can also say: We let $R$ denote a commutative ring… or Let us denote by $R$ a commutative ring…)

2. Most of them is …—> Most of them are.

3. There is a finite number of –> There are a finite number of.

(Here the quantifying expression a finite number of has the same meaning as finitely many, and it has the same syntax, i.e. it requires a plural verb).

4. The Carcdano’s formula –> Carcdano’s formula (or: The Cardano formula)

5.  The Chapter 1 –> Chapter 1.

(If a series of objects are numbered by positive integers, corresponding to ordinal numbers, no article is used: in Section 2; on page 4; in row n. However, often the numbering/labelling is not as direct and then the may apear; e.g. usually we write Definition 2.1, but you can say both inequality (2:1) and the inequality (2:1).)

6. Such map exists –> Such a map exists ( Note for every such map). see 14.

7. In the case $R$ is Noetherian –> In case $R$ is Noetherian (or: In the case where $R$ is Noetherian).

8. In case of smooth norms –> In the case of smooth norms.

9. F  is equal G –> F  is equal to G (or F equals G).

10. F is greater or equal to G –> F is greater than or equal to G.

11. Similar as A –> Similar to A.

Similarly as in Chapter 1 –> Similarly to Chapter 1. (or: As (Just as) in Chapter 1; As is the case in Chapter 1; In much the same way as in Chapter 1).

12. Disjoint with A –> Disjoint from A.

13. Let W be  the linear complement of the subspace U in V –>Let W be a linear complement of the subspace U in V.

(There are many complements of U; if you have in mind any of them, you have to use the indefinite article. On the other hand, you can say: Let W be the linear complement of the subspace U in V , described in Remark 2—here you are specifying WHICH complement you have in mind).

14. Such operator is defined by… –>Such an operator is defined by…

(The word such, when appearing before a singular countable noun, is followed by a/an. This rule does not obey if such is preceded by a quantifier: one such map; for every such map; some such difficulty).

15.  The closed sets are Borel sets –>Closed sets are Borel sets.

(The does not mean “all”. If you talk about things in general, use no article. This rule does not obey in some constructions with of —it is understood that the generality is somehow limited here:
+  The members of the collection U are called the open sets of X.
Also, use the when you are talking about a set as a whole:
+ The linear operators on V can be identified with the matrix space M)

16. The number of the solutions of (1); the set of the solutions of (1) –>The number of solutions of (1); the set of solutions of (1)

(On the other hand, you say e.g. the union of the sets $U_i$ ).

17. In 2008 Fox has shown that…–>In 2008 Fox showed that…

(If you are giving a date, it is understood that you are thinking about a definite moment in the past; you then have to use the Simple Past tense. However, you can well say, without specifying the time: Fox has shown that…—Fox proved something in the past, but when talking about it, you are also thinking about the present: IT IS PROVED NOW, because he proved it (no matter when). In such circumstances, use the Present Perfect tense).

18. This lemma allows to prove the theorem without using (2) –> This lemma allows us to prove the theorem without using (2).

(The verb allow requires an indirect object: you have to say WHOM the lemma allows to prove the theorem. If you do not want to say that it allows you (“us”), you can say: This lemma allows one to prove the theorem, that is, it allows you and the reader.

You can avoid adding us/one by using a noun or an ing-form:
+  This lemma allows proving the theorem without the use of (2),
or the passive voice:
+  This lemma allows the theorem to be proved without using (2).
The same problem concerns the verbs enable and permit. Here are examples of their correct use:
+ Repeated application of Lemma 2 enables us to write…
+  Theorem 3 enables discontinuous derivations to be built.
+  This will permit us to demonstrate that…
+ Formula (6) permits transfer of the results in Section 2 to sums of i.i.d. variables.
Another verb requiring an indirect object is remind, see 19).

19. The purpose of this section is to remind some results on… –>The purpose of this section is to remind the reader of some results on…

(If you do not want to involve the reader, you can use recall:
+ The purpose of this section is to recall some results on…).

20. We should avoid to use (2) here, because… –>We should avoid using (2) here, because…

(After some verbs you cannot use an infinitive; they have to be followed by an ingform. These include avoid, but also finish and suggest:
+ This suggests investigating the solutions of…

Here are other similar examples:
+ Section 3 is devoted to prove this theorem –>  Section 3 is devoted to proving this theorem.
+ The possibility to obtain a better bound –>  The possibility of obtaining a better bound).