I don't believe Texas has an estate tax.  Since you reference 1041, I'm assuming you are referring to the federal estate tax form for a Texas resident?  

You will need to ask specific questions to get answers here.  If you need help because you don't understand estate tax you should refer the client to someone else.  

I thought that I'm very clear on my question. My question is: Do I need to file state form 1041 (ie. TX-1041) in the state of Texas for an estate established in Texas ?

Definitely, everyone has to file federal 1041, no mater which state the estate is established.

Thank you so much for your answer.

Apparently, your question was not clear enough to @nancy2 .

An estate MAY have to file (a) an estate tax return (Form 706) and (b) annual fiduciary income tax return (Form 1041). 

Still, Form 1041 is NOT definitely (as you stated) required.  Read the 1041 instructions on "Who Must File".

@nancy2  interpreted your question as one about the TX version of Form 706.     

I don't have any TX estate or trust, so, I don't have a definite answer for you.  I like to clear my head before diving into work, and I'm helping you to get this question on Page one before it gets buried.  On April 9, only the lucky few would go past Page 2 in this forum.

In my e-library, I have a pdf file titled 

Residency and Source Income Factors for State Income Taxation of Irrevocable Non-Grantor Trusts

It's a list for state tax issues for trusts for many (all?) states. 

For TX it has one line:

Texas (0%)  No income tax imposed on trusts.

Perhaps a TX practitioner could verify no TX fiduciary income tax form is needed for a TX estate. 

Thank you for all your answers. My question is about Texas form 1041. Not federal 1041, not 706. Form 1041 is about INCOME TAX not estate tax. I stated right on the title of my question: TX state form 1041.Not sure why there is a confusion.

Any way I got the answer that I don't need to file state form 1041 in TX for an estate established in TX.