:salam2:
Sister Hayat, I think what you're asking about is the Islamic view of making promises/oaths (saying/promising in Allah (SWT)'s name that would do/not do something), right?
And as for your question:
....if in one of the requests of the wife/husband there is to not get married after his/her death,should he/she respect the request?
Then yes, it should be respected.
Unless the husband or wife fears Fitnah (like if the husband feels that by NOT getting married to another women (after his wife dies), he will end up falling in-love with another girl or he will end up committing Zina and such).
Or in the wife's case, she feels that by not getting re-married, she will not be able to feed/raise her children (and she can't receive Zakaat to support them).
Another situation (in which the promise can be broken) is when the lonliness (the feeling of being alone and that you wish you had someone special to love you) gets unbearable (you feel that you can't take it anymore) to the point that you're always sad or your Ibadah/Deen starts to go down. For example, if you/he/she always feels sad and stops talking to their friends/family, or if the husband/wife can't concentrate in their Salat, or if the person can't try to be a better Muslim because he/she needs a husband/wife to encourage them to try to be a better Muslim.
That's why Rasulallah (SAW) remarried after Khadijah (RA) died, and the same goes to Ali (RA) (and as far as I remember, Ali (RA) cried when he found out that Fatimah (RA) was going to die).