Hi Teachers

I think the genitive subject I learned was this :

USE --'s--- for the persons and animals

USE --of--- for the unliving things or the others except persons or animals

So why I sometimes see these type of genitives....

A Japanese railway company has started testing what it says is the world's fastest train......................

we can increase the examples,,,,are there any rules for this situation

OR is this the famous one in whichthere are always something against the grammar rule?
"world" can be used in the genitive case, as well as the names of countries.
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Your understanding is generally correct, redkiddy.

But more often than not there are exceptions in general rules.

Let me add some more to pieannes's list.

1. names of planets: Earth's, Saturn's, Pluto's, .....

2. time-related words: today's, yesterday's, tomorrow's, a day's work, two weeks' vacation....

I think there will be more than this.

I have,however, seen many cases where even the general rules are crumbling. Examples are: my watch's hourly chime; the road's edge; the river's level; .... But I'd like to recommend that in exam situations you play safe by following the general rules.
The genistive 's can be used for

people: John's car

organisations: the administration's agenda

time phrases: a week's pay, in a year's time

astrological bodies: the sun's rays

Your examples of non-people, non-animal ['s] fall under one of the other 3 sub-headings

Hope this helps

Kind regards