Ég gleymdi reglustikunni heima.

Because the verb gleyma normally takes the dative case in Icelandic.

So:

• að gleyma einhverju = to forget something
• reglustikunni is the dative singular definite form of reglustika.

A useful comparison:

• reglustika = a ruler
• reglustikuna = the ruler (accusative)
• reglustikunni = the ruler (dative)

With gleyma, Icelandic wants the dative form, so reglustikunni is the correct choice.

In Icelandic, the definite article is very often attached to the end of the noun instead of being a separate word.

So reglustikunni includes both:

• the noun reglustiku-
• and the idea of the

In other words, it is roughly ruler-the, but also marked for case.

This is very common in Icelandic. English uses a separate word (the), but Icelandic often uses a suffix instead.

Gleymdi is the past tense of gleyma (to forget).

So:

• að gleyma = to forget
• ég gleymi = I forget / I am forgetting
• ég gleymdi = I forgot

Also, Icelandic does not need an extra word like did here. English says I forgot, and Icelandic simply says ég gleymdi.

Usually, no. Icelandic normally keeps the subject pronoun in a full sentence.

So Ég gleymdi... is the normal way to say I forgot...

One reason is that gleymdi by itself does not clearly show I only; the same past-tense form can also be used with he/she/it. Because of that, ég helps make the subject clear.

Heima means at home.

It is an adverb, not a noun with a preposition. So Icelandic does not need a separate word for at here.

Compare:

• heima = at home
• heim = home, to home, homeward

So:

• Ég er heima = I am at home
• Ég fer heim = I am going home

In your sentence, heima means the ruler was forgotten at home.

In normal use, it usually means I left the ruler at home / I forgot it there.

So the natural idea is:

• the ruler is still at home
• I do not have it with me

Grammatically, the sentence could be understood as I forgot the ruler at home, but in everyday meaning the stronger implication is usually I accidentally left it at home.

This is a very normal Icelandic word order:

• Ég = subject
• gleymdi = verb
• reglustikunni = object
• heima = adverb of place

So the pattern is basically:

subject + verb + object + place

That makes the sentence sound natural and neutral. Icelandic word order can be moved around for emphasis, but this version is the straightforward one.

Reglustika is a feminine noun.

That matters because gender affects the endings the noun takes. In this sentence, the feminine noun changes form because it is:

• singular
• definite
• dative

That is why you get reglustikunni.

A few useful forms are:

• reglustika = a ruler
• reglustiku = a ruler (object/dative form)
• reglustikunni = the ruler (dative)

You would remove the definite ending and use the indefinite form required by the verb.

Since gleyma takes the dative, you would say:

Ég gleymdi reglustiku heima.

That means I forgot a ruler at home.

So the difference is:

• reglustiku = a ruler
• reglustikunni = the ruler

Yes. Reglustika is the normal Icelandic word for a ruler in the measuring-tool sense.

So if you mean the straight object used in school or for measuring, reglustika is the right word.